Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups

Abstract

The notion of matched pair $(F,G,▹,◃)$ of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group $F\bowtie G$. We find in this paper some sufficient conditions when equations ${\nu }_m(F\bowtie G)={\nu }_m\left(F\right){\nu }_m\left(G\right)$ for all $m\ge 0$ imply that $F\bowtie G$ is the external direct product of $F\times G$, where ${\nu }_m$ denotes m-th indicator of the regular representation of a finite group. A comparison with indicators of bismash product Hopf algebras ${\mathbb{C}}^G\#\mathbb{C}F$ is also mentioned.

1. Introduction

In the representation theory of finite groups, Frobenius–Schur indicators ${\nu }_2$ arose as some scalars providing an important criteria about whether an irreducible (complex) representation of a finite group is self-dual up to isomorphisms ([1]). This notion is now often regarded as a particular case of higher Frobenius–Schur indicators (referred to as indicators for short) are denoted by ${\nu }_m(m\ge 0)$ and also defined via characters. For any natural number m, the m-th indicator of a finite-dimensional representation V for a finite group G is

$${\nu }_m\left(V\right):={\displaystyle \frac{1}{\left|G\right|}}\sum _{g\in G}{\chi }_V\left(g^m\right),$$

where ${\chi }_V$ is the character of V. In fact, the values of the indicators derive from the root number functions in G (e.g., [2] Lemma 4.4). We remark that the notion of indicators can be defined for representations of semisimple (quasi-)Hopf algebras ([3,4,5], etc.) and objects in spherical fusion categories ([6,7], etc.), where the indicators are invariant under certain equivalences.

Although the values of indicators (of all the representations) will contain a large amount of information for a finite group G, some properties can be obtained by just considering the indicators ${\nu }_m\left(G\right)$ of the regular representation $\mathbb{C}G$. For example, it is known that $exp\left(G\right)$ is the period of the sequence ${\left\{{\nu }_m\left(G\right)\right\}}_{m\ge 0}$. In fact, there is an analog for indicators of the regular representation to the case of non-semisimple Hopf algebras ([8,9]).

This paper is a discussion of the values of ${\nu }_m\left(L\right)$, when the finite group L is factorizable in the sense that $L=FG$ for subgroups F and G satisfying $F\cap G=1$. Specifically, we attempt to find sufficient conditions for when

$${\nu }_m\left(L\right)={\nu }_m\left(F\right){\nu }_m\left(G\right)(\forall m\ge 0)$$

implies that L is an internal direct product of F and G, as the converse holds due to a direct observation; ${\nu }_m\left(L\right)$ equals the number of elements in L with orders dividing m.

Our basic method is that L is known to be canonically isomorphic to a group denoted by $F\bowtie G$. It is determined by a kind of matched pair$(F,G,▹,◃)$ of groups, where ▹ and ◃ are group actions between F and G with certain requirements. This notion was introduced by Takeuchi [10] for constructions of particular Hopf algebras, and is a generalization of external direct products and semi-direct products. With such terms, we prove in this paper the following result, which is a combination of Proposition 2 and Theorem 2:

Theorem 1. 

Let $(F,G,▹,◃)$ be a matched pair of finite groups. Suppose that

$${\nu }_m(F\bowtie G)={\nu }_m\left(F\right){\nu }_m\left(G\right)(\forall m\ge 0).$$

(1)

Then, $F\bowtie G$ is the external direct product $F\times G$, if either of the following conditions holds:

(1) 

$gcd\left(\right|F|,|G\left|\right)=1$;

(2) 

F and G are nilpotent, and $gcd\left(\right|F|,|G\left|\right)=p$ is a prime number. Moreover, p, ${\displaystyle \frac{\left|F\right|}{p}}$ and ${\displaystyle \frac{\left|G\right|}{p}}$ are coprime to each other.

However, we can easily find counter-examples where the assumptions in Theorem 1 are not satisfied, namely, non-direct products $F\bowtie G$ with condition (1). Furthermore, we are interested in necessary and sufficient conditions for when (1) implies $F\bowtie G=F\times G$.

The organization of this paper is as follows. Section 2 is a collection of preliminaries on the various (internal and external) products of groups and their relations, as well as higher Frobenius–Schur indicators of finite groups. In Section 3, we explore some properties for stabilizers of the actions determined in matched pairs of groups, and then establish our main results stated in Theorem 1 with an elementary fact that the normality of subgroups can be obtained by equations on the indicators. Afterwards, examples are provided in order to partially explain the necessity of the assumptions in our main theorem, and indicators between matched pairs and bismash product Hopf algebras are compared.

2. Preliminaries: Products of Groups and Higher Frobenius–Schur Indicators

2.1. Factorizations, Direct and Semi-Direct Products of Groups

We begin by specifying some notions on the products of groups in the literature, including factorizations, direct and semi-direct products.

Definition 1. 

Let L be a group with subgroups F and G. Suppose

$$L=FG\mathit{and}F\cap G=1.$$

Then, we say that $L=FG$ is a factorization into subgroups (or F can be factorized into subgroups F and G).

In particular, we recall also the definition of internal direct product:

Definition 2. 

A group L is called the internal direct product of F and G, if all the following conditions hold:

(1) 

$L=FG$ and $F\cap G=1$;

(2) 

$ax=xa$ holds for all $a\in F$ and $x\in G$.

In this case, we also say that the factorization $L=FG$ is direct.

To avoid confusion in this paper, the direct product whose underlying set is the Cartesian product of groups is said to be external (see [11] p. 41, for example). Specifically:

Definition 3. 

Let F and G be groups. The external direct product $F\times G$ is defined to be their Cartesian product with operation as follows:

$$(a,x)(b,y):=(ab,xy)(\forall a,b\in F,\forall x,y\in G).$$

Remark 1. 

If a group L is the internal direct product of subgroups F and G, then there is a group isomorphism

$$F\times G\to L,(a,x)\mapsto ax.$$

However, by the (external) semi-direct product $F⋊G$ of groups F and G, we always mean their Cartesian product with operations

$$(a,x)(b,y):=\left(a\right(x▹b),xy)(\forall a,b\in F,\forall x,y\in G),$$

where $▹:G\times F\to F$ is a group action satisfying that $x▹-$ is an endomorphism for each $x\in G$.

2.2. Matched Pair of Groups and Factorizable Groups

The notion of matched pair of groups was introduced in [10] for the purpose of constructing certain Hopf algebras called bismash products. We recall the definition in [12,13]:

Definition 4 

([12] Definition 1.1). Let F and G be groups with group actions $▹:G\times F\to F$ and $◃:G\times F\to G$. We say that $(F,G,▹,◃)$ is a matched pair of groups, if

$$x▹ab=(x▹a)\left(\right(x◃a)▹b)\mathit{and}xy◃a=(x◃(y▹a\left)\right)(y◃a)$$

(2)

hold for all $a,b\in F$ and $x,y\in G$.

It is known ([10] Proposition 2.2) that if $(F,G,▹,◃)$ is a matched pair of groups, then the Cartesian product $F\times G$ becomes a group with identity $(1,1)$ and operation defined by:

$$(a,x)(b,y):=\left(a\right(x▹b),(x◃b\left)y\right)(\forall a,b\in F,\forall x,y\in G).$$

(3)

This group is always denoted by $F\bowtie G$.

Remark 2. 

One can find, according to Equations (2), that

$$x▹1=1\mathit{and}1◃a=1$$

(4)

hold for all $a\in F$ and $x\in G$. See ([10] Section 2). As a consequence, $F\bowtie G$ has two subgroups

$$F\bowtie 1=F\times 1\mathit{and}1\bowtie G=1\times G$$

(5)

which are indeed external direct products. Furthermore, the equations

$$(a,1)(b,y)=(ab,y)\mathit{and}(a,x)(1,y)=(a,xy)$$

(6)

hold in $F\bowtie G$ for all $a,b\in F$ and $x,y\in G$.

We remark also that some useful formulas on matched pair of groups were shown in ([14] Lemma 4.2) and ([15] Proposition 2.4).

In fact, the group $F\bowtie G$ constructed is a generalization of external direct product and semi-direct product. In order to explain, we will say that a group action $▹:G\times F\to F$ is trivial, if $x▹-:F\to F$ is the identity map for each $x\in G$. Similarly, $◃:G\times F\to G$ is trivial if $-◃a:G\to G$ is the identity map for each $a\in F$.

Lemma 1. 

Let $(F,G,▹,◃)$ be a matched pair of groups. Then, the following are equivalent:

(1) 

The group action ◃ is trivial;

(2) 

$F\bowtie G$ is the semi-direct product $F⋊G$ determined by the group action ▹;

(3) 

The subgroup $F\times 1$ is normal in $F\bowtie G$.

Proof. 

(1)⇒(2): Suppose $a,b\in F$ and $x,y\in G$. The triviality of ◃ implies that $x◃a=x$, and hence,

$$x▹ab\stackrel{\left(2\right)}{=}(x▹a)((x◃a)▹b)=(x▹a)(x▹b).$$

We concluded that $x▹-$ is an endomorphism of the group F.

Moreover, $x◃b=x$ holds as well. Thus, we know by (3) that

$$(a,x)(b,y)=\left(a\right(x▹b),(x◃b\left)y\right)=\left(a\right(x▹b),xy)$$

holds in $F\bowtie G$, which coincides with the operation on the semi-direct product $F⋊G$.

(2)⇒(3): Suppose $a,b\in F$ and $x\in G$. Evidently ${(a,x)}^{-1}=(x^{-1}▹a^{-1},x^{-1})$ holds in the semi-direct product $F⋊G$. The normality of the subgroup $F\times 1$ in $F⋊G$ is obtained by the following computations:

$$\begin{array}{ccc}\hfill (a,x)(b,1){(a,x)}^{-1}& =& (a(x▹b),x)(x^{-1}▹a^{-1},x^{-1})=\left(a(x▹b)\left(x▹(x^{-1}▹a^{-1})\right),x{x}^{-1}\right)\hfill \\ & =& (a(x▹b)a^{-1},1)\in F\times 1.\hfill \end{array}$$

(3)⇒(1): For all $a\in F$ and $x\in G$, we compute in $F\bowtie G$ that

$$(1,x)(a,1){(1,x)}^{-1}\stackrel{\left(5\right)}{=}(1,x)(a,1)(1,x^{-1})\stackrel{\left(6\right)}{=}(1,x)(a,x^{-1})\stackrel{\left(3\right)}{=}(x▹a,(x◃a)x^{-1}).$$

However, $(1,x)(a,1){(1,x)}^{-1}\in F\times 1$ due to the normality. One can conclude that for all $a\in F$ and $x\in G$,

$$(x◃a)x^{-1}=1,$$

which is equivalent to $x◃a=x$. As a conclusion, the action ◃ is trivial. □

Lemma 2. 

Let $(F,G,▹,◃)$ be a matched pair of groups. Then, the following are equivalent:

(1) 

The group actions ▹ and ◃ are both trivial;

(2) 

$F\bowtie G$ is the external direct product $F\times G$;

(3) 

The subgroups $F\times 1$ and $1\times G$ are both normal in $F\bowtie G$.

Proof. 

(1)⇒(2): Suppose $a,b\in F$ and $x,y\in G$. The triviality of ◃ and ▹ follow that $x◃b=x$ and $x◃b=x$. Thus, we know by (3) that

$$(a,x)(b,y)=\left(a\right(x▹b),(x◃b\left)y\right)=(ab,xy)$$

holds in $F\bowtie G$, which coincides with the operation on the external direct product $F\times G$.

(2)⇒(3): This is clear.

(3)⇒(1): By Lemma 1, ◃ is trivial if the subgroup $F\times 1$ is normal in $F\bowtie G$. A similar argument provides that the normality of $1\times G$ implies that ▹ will be trivial as well. □

Remark 3. 

There exist examples of matched pairs of groups with non-trivial group actions, such as ([12] Example 1.3): $S_n\cong C_n\bowtie S_{n-1}$ when $n\ge 4$.

In fact, a matched pair of groups is equivalent to a factorization into two subgroups in the sense of Definition 1.

Lemma 3 

([10] Proposition 2.4 or [16] Section 3). Suppose a group L can be factorized into subgroup F and G. Then, there exists a unique matched pair $(F,G,▹,◃)$ such that

$$xa=(x▹a)(x◃a)(\forall a\in F,\forall x\in G),$$

or equivalently,

$$F\bowtie G\to L,(a,x)\mapsto ax$$

is a group isomorphism.

2.3. Higher Frobenius–Schur Indicators

Let G be a finite group. Suppose G represents a finite-dimensional complex vector space V with character ${\chi }_V$. For each natural number m, the m-th (Frobenius–Schur) indicator of representation V is defined as the scalar

$${\nu }_m\left(V\right):={\displaystyle \frac{1}{\left|G\right|}}\sum _{g\in G}{\chi }_V\left(g^m\right)\in \mathbb{C}$$

See ([2] Lemma 4.4) for the backgrounds on higher indicators.

We aim to study relations between the indicators of finite groups F, G and $F\bowtie G$. In particular, it is easy to find an evident relation when $F\bowtie G$ is the external direct product $F\times G$:

Lemma 4. 

Let F and G be finite groups. Suppose F represents a finite-dimensional complex vector space V with character ${\chi }_V$, and G represents a finite-dimensional complex vector space W with character ${\chi }_W$. Then,

$${\nu }_m(V\otimes W)={\nu }_m\left(V\right){\nu }_m\left(W\right)(\forall m\ge 0),$$

(7)

where the tensor product $V\otimes W$ over $\mathbb{C}$ is regarded as the representation space of the external direct product $F\times G$ defined by

$$(a,x)·(v\otimes w):=av\otimes xw(\forall a\in F,x\in G,\forall v\in V,w\in W).$$

Proof. 

This can be shown by direct computations according to ([17] Section 3.2), for example. □

In this paper, we will concentrate on higher indicators of regular representations, which can be obtained by counting elements with certain orders. To state this lemma, denote the m-th indicator of the regular representation of a finite group G by ${\nu }_m\left(G\right)$ for simplicity:

Lemma 5. 

Let G be a finite group. Then ${\nu }_m\left(G\right)=\left|\{x\in G\mid x^m=1\}\right|$ for all $m\ge 0$.

Proof. 

This is immediate by the fundamental fact that

$${\chi }_G\left(x\right)=\left\{\begin{array}{cc}\left|G\right|,\hfill & \mathrm{when}x=1,\hfill \\ 0,\hfill & \mathrm{when}x\ne 1,\hfill \end{array}\right.$$

where ${\chi }_G$ is the character of the regular representation of G. □

Remark 4. 

Isomorphic groups have identical indicators of the regular representations.

Corollary 1. 

${\nu }_m\left(G\right)=1$ when the natural number m is coprime to $\left|G\right|$.

Note in Remark 1 that internal and external direct products are isomorphic. Thus, it is straightforward to find that:

Proposition 1. 

With notations in the paragraph before Lemma 5, we have

$${\nu }_m(F\times G)={\nu }_m\left(F\right){\nu }_m\left(G\right)(\forall m\ge 0),$$

(8)

where F and G are finite groups.

On the other hand, suppose that a factorization of a finite group $L=FG$ is direct. Then, a similar equation ${\nu }_m\left(L\right)={\nu }_m\left(F\right){\nu }_m\left(G\right)$ holds for any $m\ge 0$.

3. Sufficient Conditions When a Factorization Becomes Direct via Indicators

This section is devoted to investigating sufficient conditions for $F\bowtie G$ being the external direct product $F\times G$, when a similar relation to (8) on indicators of F, G, and $F\bowtie G$ holds. Specifically:

Question 1. 

Suppose $(F,G,⊳,⊲)$ is a matched pair of finite groups satisfying

$${\nu }_m(F\bowtie G)={\nu }_m\left(F\right){\nu }_m\left(G\right)(\forall m\ge 0).$$

(9)

When is $F\bowtie G$ the direct product $F\times G$?

3.1. For Matched Pair of Groups with Relatively Prime Orders

In order to search for positive answers to Question 1, the following lemma on the normality of subgroups via the language of indicators will be useful.

Lemma 6. 

Suppose G is a subgroup of a finite group L. If ${\nu }_m\left(L\right)={\nu }_m\left(G\right)$ holds for some multiple m of $\left|G\right|$, then G is normal in L.

Proof. 

It is clear by Lemma 5 that ${\nu }_m\left(G\right)=\left|G\right|$ if the order $\left|G\right|$ divides m. Furthermore, the assumption ${\nu }_m\left(L\right)={\nu }_m\left(G\right)$ follows that all the elements in L with orders dividing m belong to the subgroup G. Consequently, the subgroup G is closed under the conjugate action of L, which means that G is normal. □

With the help of Lemma 6, we can easily check the case for matched pair of groups with relatively prime orders:

Proposition 2. 

Suppose $(F,G,⊳,⊲)$ is a matched pair of finite groups satisfying that $\left|F\right|$ and $\left|G\right|$ are relatively prime. If

$${\nu }_m(F\bowtie G)={\nu }_m\left(F\right){\nu }_m\left(G\right)(m=|F|,|G\left|\right),$$

(10)

then $F\bowtie G$ is the direct product $F\times G$.

Proof. 

As $\left|F\right|$ and $\left|G\right|$ are relatively prime, we know by Corollary 1 that ${\nu }_{\left|F\right|}\left(G\right)=1$. We choose $m=\left|F\right|$ in Equation (10) and obtain

$${\nu }_{\left|F\right|}(F\bowtie G)={\nu }_{\left|F\right|}\left(F\right)={\nu }_{\left|F\right|}(F\times 1).$$

Thus, by Lemma 6, the subgroup $F\times 1$ is normal in $F\bowtie G$. In other words, the action ◃ is trivial according to Lemma 1.

Similarly, we can choose $m=\left|G\right|$ in Equation (10) to find that ▹ is also trivial. As a conclusion, it follows by Lemma 2 that $F\bowtie G=F\times G$. □

Remark 5. 

According to Lemma 3, we can also obtain an internal version of Proposition 2. Suppose a group L can be factorized into subgroups F and G satisfying that $\left|F\right|$ and $\left|G\right|$ are relatively prime. If

$${\nu }_m\left(L\right)={\nu }_m\left(F\right){\nu }_m\left(G\right)(m=|F|,|G\left|\right),$$

then the factorization $L=FG$ is direct.

3.2. For Matched Pair of Groups with Orders Not Relatively Prime

Let us begin by considering matched pair of groups which include cyclic groups of prime orders.

For the remainder of this paper, we always denote by $C_n$ the cyclic group of order n.

Lemma 7. 

Let $(F,G,▹,◃)$ be a matched pair of finite groups. Suppose $x\in G$ is an element of prime order p. Then, p divides the cardinal number of $\{a\in F\mid x▹a\ne a\}$.

Proof. 

Note that $x^p▹a=1▹a=a$ holds for all $a\in G$. Consequently, $x▹-$ is a product of disjoint p-cycles on F, as it is a permutation with order dividing p. It follows that p divides the number of elements moved by $x▹-$. Or equivalently, p divides the cardinal number of $\{a\in F\mid x▹a\ne a\}$. □

Corollary 2. 

Let $(F,C_p,▹,◃)$ be a matched pair of finite groups, where $C_p=\langle x\rangle$ is the cyclic group of prime order p. Suppose $p\mid \left|F\right|$. Then,

$$\{a\in F\mid x▹a=a\}$$

(11)

is a subgroup of F whose order is a multiple of p.

Proof. 

Since x is a generator of the cyclic group $C_p$, the subset (11) equals

$$F^{C_p}:=\{a\in F\mid \forall y\in C_p,y▹a=a\}.$$

According to ([14] Lemma 4.1), it is a subgroup of F, whose order will be

$$\left|F\right|-\left|\right\{a\in F\mid x▹a\ne a\left\}\right|,$$

a multiple of p as well due to Lemma 7. □

Particularly for a matched pair of form $(C_p,C_p,▹,◃)$, we know by Corollary 2 that the action ▹ will be trivial. One can find by a similar argument that ◃ is also trivial. Therefore, the following corollary is obtained as a result of Lemma 2:

Corollary 3. 

Suppose p is a prime number. Then, $C_p\bowtie C_p=C_p\times C_p$.

Remark 6. 

This can also be shown by the fact that groups of order $p^2$ are all abelian.

Our main result in this subsection is as follows, which can be a partial answer to Question 1 for nilpotent matched pair of groups with related orders:

Theorem 2. 

Let $(F,G,▹,◃)$ be a matched pair of finite nilpotent groups. Suppose that $gcd\left(\right|F|,|G\left|\right)=p$ is a prime number, and that p, ${\displaystyle \frac{\left|F\right|}{p}}$ and ${\displaystyle \frac{\left|G\right|}{p}}$ are coprime to each other. If

$${\nu }_m(F\bowtie G)={\nu }_m\left(F\right){\nu }_m\left(G\right)(\forall m\ge 0),$$

(12)

then $F\bowtie G$ is the external direct product $F\times G$.

Proof. 

At first, the assumptions on the orders imply that the Sylow p-subgroups of F and G both have order p. Note by ([11] Theorem 5.39), for example, that any nilpotent group is the internal direct product of Sylow subgroups. Thus, there are direct factorizations

$$F=C_p{F}^{\prime }\mathrm{and}G=C_p{G}^{\prime }$$

(13)

into subgroups, where $C_p$ denotes cyclic subgroups of F and G without confusions.

Meanwhile, since $|F^{\prime }|={\displaystyle \frac{\left|F\right|}{p}}$ is coprime to $\left|G\right|$ and p, one can find by Corollary 1 that ${\nu }_{|F^{\prime }|}\left(G\right)={\nu }_{|F^{\prime }|}\left(C_p\right)=1$ holds. Therefore, we compute

$$\begin{array}{ccc}\hfill {\nu }_{|F^{\prime }|}(F\bowtie G)& \stackrel{\left(12\right)}{=}& {\nu }_{|F^{\prime }|}\left(F\right){\nu }_{|F^{\prime }|}\left(G\right)={\nu }_{|F^{\prime }|}\left(F\right)\hfill \\ & \stackrel{\mathrm{Proposition} 1}{=}& {\nu }_{|F^{\prime }|}\left(C_p\right){\nu }_{|F^{\prime }|}\left(F^{\prime }\right)={\nu }_{|F^{\prime }|}\left(F^{\prime }\right)={\nu }_{|F^{\prime }|}(F^{\prime }\times 1).\hfill \end{array}$$

Then, it follows by Lemma 6 that the subgroup $F^{\prime }\times 1$ is normal in $F\bowtie G$. Similarly, the subgroup $1\times G^{\prime }$ is also normal in $F\bowtie G$.

Now, we conclude that the product $F^{\prime }\bowtie G^{\prime }=(F^{\prime }\times 1)(1\times G^{\prime })$ of normal subgroups will be a normal subgroup of $F\bowtie G$ as well, and furthermore,

$$F^{\prime }\bowtie G^{\prime }\stackrel{\mathrm{Lemma} 2}{=}{F}^{\prime }\times G^{\prime }=(1\times G^{\prime })(F^{\prime }\times 1)$$

(14)

is direct.

On the other hand, the normality of $F^{\prime }\times 1$ implies also that the product $F^{\prime }\bowtie G=(F^{\prime }\times 1)(1\times G)$ of subgroups is again a subgroup of $F\bowtie G$, and that

$$F^{\prime }\bowtie G\stackrel{\mathrm{Lemma} 1}{=}{F}^{\prime }⋊G.$$

(15)

However, it is known that the definition of semi-direct product is:

$$F^{\prime }⋊G=(1\times G)(F^{\prime }\times 1)$$

(16)

into subgroups, which can be obtained by the argument:

$$\forall a\in F^{\prime },\forall x\in G,(a,x)=(1,x)(x^{-1}▹a,1)\in F^{\prime }⋊G.$$

In fact, another factorization of $F^{\prime }⋊G$ can be further induced:

$$\begin{array}{ccc}\hfill F^{\prime }⋊G& \stackrel{\left(16\right)}{=}& (1\times G)(F^{\prime }\times 1)\stackrel{\left(13\right)}{=}(1\times C_p{G}^{\prime })(F^{\prime }\times 1)\hfill \\ & =& (1\times C_p)(1\times G^{\prime })(F^{\prime }\times 1)\stackrel{\left(14\right)}{=}(1\times C_p)(F^{\prime }\times G^{\prime }).\hfill \end{array}$$

(17)

As a conclusion, we proceed to compute that

$$\begin{array}{ccc}& & F\bowtie G\stackrel{\left(13\right)}{=}{C}_p{F}^{\prime }\bowtie G=(C_p\times 1)(F^{\prime }\bowtie G)\stackrel{\left(15\right)}{=}(C_p\times 1)(F^{\prime }⋊G)\hfill \\ & \stackrel{\left(17\right)}{=}& (C_p\times 1)(1\times C_p)(F^{\prime }\times G^{\prime })=(C_p\bowtie C_p)(F^{\prime }\times G^{\prime })\stackrel{\mathrm{Corollary} 3}{=}(C_p\times C_p)(F^{\prime }\times G^{\prime }).\hfill \end{array}$$

This is indeed a factorization of $F\bowtie G$ into subgroups, since $(C_p\times C_p)\cap (F^{\prime }\times G^{\prime })=\left\{(1,1)\right\}$.

Finally, let us verify that ${\nu }_m(F\bowtie G)={\nu }_m(C_p\times C_p){\nu }_m(F^{\prime }\times G^{\prime })$ holds for all $m\ge 0$. Specifically:

$$\begin{array}{ccc}\hfill {\nu }_m(F\bowtie G)& \stackrel{\left(12\right)}{=}& {\nu }_m\left(F\right){\nu }_m\left(G\right)\stackrel{\mathrm{Proposition} 1}{=}{\nu }_m\left(C_p\right){\nu }_m\left(F^{\prime }\right){\nu }_m\left(C_p\right){\nu }_m\left(G^{\prime }\right)\hfill \\ & =& {\nu }_m\left(C_p\right){\nu }_m\left(C_p\right){\nu }_m\left(F^{\prime }\right){\nu }_m\left(G^{\prime }\right)\stackrel{\mathrm{Proposition} 1}{=}{\nu }_m(C_p\times C_p){\nu }_m(F^{\prime }\times G^{\prime })\hfill \end{array}$$

Consequently, we know by Remark 5 that the factorization $F\bowtie G=(C_p\times C_p)(F^{\prime }\times G^{\prime })$ is direct, which implies that the subgroups

$$F\times 1=(C_p\times 1)(F^{\prime }\times 1)\mathrm{and}1\times G=(1\times C_p)(1\times G^{\prime })$$

of $F\bowtie G$ are both normal, and hence, $F\bowtie G=F\times G$ due to Lemma 2. □

3.3. Examples When the Assumptions Are Not Satisfied

It is not hard to find some matched pairs $(F,G,▹,◃)$ of finite groups satisfying Equation (9), but $F\bowtie G$ is not direct. In order to determine the regular indicators of a specific finite group, the following known lemma should be noted:

Lemma 8. 

The exponent $exp\left(G\right)$ of a finite group G is equal to the least positive period of the sequence ${\left\{{\nu }_m\left(G\right)\right\}}_{m\ge 0}$ of regular indicators.

Remark 7. 

According to the lemma above and Corollary 1, we conclude that the regular indicators of a finite group G are completely determined by the values ${\nu }_m\left(G\right)$ for those m being positive divisors of $\left|G\right|$ except 1 and $\left|G\right|$.

Let us provide a particular class of examples where the greatest common divisor of $\left|F\right|$ and $\left|G\right|$ is an odd prime number:

Example 1. 

Let p be an odd prime and $1\le k\le p-1$. Suppose that $C_{p^2}=\langle a\rangle$ and $C_p=\langle x\rangle$ are cyclic groups of order $p^2$ and p, respectively. Then, the group action ${▹}_k:C_p\times C_{p^2}\to C_p$ defined by

$$x{▹}_{k}a=a^{kp+1}$$

(18)

is a non-trivial group action, which determines a semi-direct product $C_{p^2}{⋊}_k{C}_p$.

The table of regular indicators of groups $C_{p^2}{⋊}_k{C}_p$, $C_{p^2}$ and $C_p$ is as follows:

$$\begin{array}{cccc}m& {\nu }_m\left(C_{p^2}{⋊}_k{C}_p\right)& {\nu }_m\left(C_{p^2}\right)& {\nu }_m\left(C_p\right)\\ p& p^2& p& p\\ p^2& p^3& p^2& p\end{array}$$

Thus, $C_{p^2}{⋊}_k{C}_p$ satisfies (9), but it is not an external direct product.

Proof. 

It is straightforward to verify that $x▹-:C_{p^2}\to C_{p^2}$ is a group automorphism. Then, one can compute for all $0\le i\le p^2$ and $0\le j\le p-1$ that

$$\begin{array}{ccc}& & x^j{▹}_{k}a=x^{j-1}{▹}_k\left(x{▹}_{k}a\right)\stackrel{\left(18\right)}{=}{x}^{j-1}{▹}_k{a}^{kp+1}=x^{j-2}{▹}_k\left(x{▹}_k{a}^{kp+1}\right)\hfill \\ & =& x^{j-2}{▹}_k{\left(x{▹}_{k}a\right)}^{kp+1}\stackrel{\left(18\right)}{=}{x}^{j-2}{▹}_k{\left(a^{kp+1}\right)}^{kp+1}=x^{j-2}{▹}_k{a}^{{(kp+1)}^2}=\cdots =a^{{(kp+1)}^j},\hfill \end{array}$$

and hence,

$$x^j{▹}_k{a}^i={\left(x^j{▹}_{k}a\right)}^i={\left(a^{{(kp+1)}^j}\right)}^i=a^{i{(kp+1)}^j}.$$

(19)

Furthermore, we compute powers ${(a^i,x^j)}^m$ in $C_{p^2}{⋊}_k{C}_p$ for all $m\ge 0$:

• When $j=0$: ${(a^i,x^j)}^m=(a^{mi},1);$
• When $1\le j\le p-1$:

  $$\begin{array}{ccc}\hfill {(a^i,x^j)}^m& =& (a^i,x^j)(a^i,x^j){(a^i,x^j)}^{m-2}\stackrel{\left(19\right)}{=}(a^i\left(x^j{▹}_k{a}^i\right),x^{2j}){(a^i,x^j)}^{m-2}\hfill \\ & =& (a^i{a}^{i{(kp+1)}^j},x^{2j}){(a^i,x^j)}^{m-2}=(a^{i[1+{(kp+1)}^j]},x^{2j}){(a^i,x^j)}^{m-2}\hfill \\ & =& \cdots =(a^{i[1+{(kp+1)}^j+\cdots +{(kp+1)}^{(m-1)j}]},x^{mj})=\left(a^{i{\displaystyle \frac{{(kp+1)}^{mj}-1}{kp}}},x^{mj}\right).\hfill \end{array}$$

According to Remark 7, it is sufficient to consider the case when m is p and $p^2$ in order to determine the regular indicators ${\nu }_m\left(C_{p^2}{⋊}_k{C}_p\right)$. Note at first that

$${(kp+1)}^{mj}-1=\sum _{t=1}^{mj}\left({\displaystyle \genfrac{}{}{0pt}{}{mj}{t}}\right)k^t{p}^t\equiv mjkp+{\displaystyle \frac{j(mj-1)}{2}}m{k}^2{p}^2\equiv \left\{\begin{array}{cc}jk{p}^2,\hfill & \mathrm{if}m=p\hfill \\ 0,\hfill & \mathrm{if}m=p^2\hfill \end{array}\right.\left(\mathrm{mod}{p}^3\right).$$

On the other hand, as $x^{mj}=1$ holds when $m=p$ and $p^2$, and hence,

$$\begin{array}{ccc}\hfill (a^{i{\displaystyle \frac{{(kp+1)}^{mj}-1}{kp}}},x^{mj})=(1,1)& ⟺& a^{i{\displaystyle \frac{{(kp+1)}^{mj}-1}{kp}}}=1⟺p^2\mid i{\displaystyle \frac{{(kp+1)}^{jm}-1}{kp}}\hfill \\ & ⟺& p^3\mid i[{(kp+1)}^{jm}-1].\hfill \end{array}$$

We conclude that every element of $C_{p^2}{⋊}_k{C}_p$ has order dividing $p^2$, and that

$$\begin{array}{ccc}\hfill {(a^i,x^j)}^p=(1,1)& ⟺& 1\le j\le p-1,p^3\mid ijk{p}^2\mathrm{or}j=0,p^2\mid pi\hfill \\ & ⟺& 1\le j\le p-1,p\mid ijk\mathrm{or}j=0,p\mid i\hfill \\ & ⟺& p\mid i.\hfill \end{array}$$

Therefore, one can obtain the value of indicators by Lemma 5 that

$${\nu }_p\left(C_{p^2}{⋊}_k{C}_p\right)=p^2={\nu }_p\left(C_{p^2}\right){\nu }_p\left(C_p\right)\mathrm{and}{\nu }_{p^2}\left(C_{p^2}{⋊}_k{C}_p\right)=p^3={\nu }_{p^2}\left(C_{p^2}\right){\nu }_{p^2}\left(C_p\right).$$

□

There are also examples when F and G are groups with even orders. We provide the simplest one as follows but omit the details of the proof:

Example 2. 

Let $C_8=\langle a\rangle$ and $C_2=\langle x\rangle$ be cyclic groups of order 8 and 2, respectively. Define the group actions ${▹}_2$ of $C_2$ on $C_8$ by $x{▹}_{2}a=a^5,$ which determines the semi-direct products $C_8{⋊}_2{C}_2$. The table of indicators of $C_8{⋊}_2{C}_2$, $C_8$ and $C_2$ is as follows:

$$\begin{array}{cccc}m& {\nu }_m\left(C_8{⋊}_2{C}_2\right)& {\nu }_m\left(C_8\right)& {\nu }_m\left(C_2\right)\\ 2& 4& 2& 2\\ 4& 8& 4& 2\\ 8& 16& 8& 2\end{array}.$$

Thus, $C_8{⋊}_2{C}_2$ satisfies (9). However, it is not an external direct product.

3.4. Comparison with Regular Indicators of Bismash Product Hopf Algebras

As we have mentioned in the first paragraph of Section 2.2, there are two semisimple complex Hopf algebras called bismash products constructed in [10] from matched pair $(F,G,▹,◃)$ of finite groups. They are usually denoted by ${\mathbb{C}}^G\#\mathbb{C}F$ and $\mathbb{C}G\#{\mathbb{C}}^F$ in the literature (c.f. ([18] Section 1) and ([12] Section 1)), whose indicators are related with $F\bowtie G$ by an external version of ([8] Corollary 5.7):

Lemma 9. 

Let $(F,G,▹,◃)$ be a matched pair of finite groups. Then, for all $m\ge 0$,

$${\nu }_m({\mathbb{C}}^G\#\mathbb{C}F)={\nu }_m(\mathbb{C}G\#{\mathbb{C}}^F)={\nu }_m(F\bowtie G).$$

Remark 8. 

The definition of higher regular indicators ${\nu }_m\left(H\right)$ of a (semisimple) Hopf algebra H is found in ([4] Section 2.1) and ([8] Definition 2.1), which equals ${\nu }_m\left(L\right)$ if $H=\mathbb{C}L$ is a group algebra of a finite group L.

According to the structure of the bismash product Hopf algebra ${\mathbb{C}}^G\#\mathbb{C}F$ (see ([4] Section 3), for example), the following corollary is a consequence of Proposition 2 and Theorem 2:

Corollary 4. 

Let $(F,G,▹,◃)$ be a matched pair of finite groups. Suppose either of the following conditions holds:

(1) 

$gcd\left(\right|F|,|G\left|\right)=1$;

(2) 

F and G are nilpotent, and $gcd\left(\right|F|,|G\left|\right)=p$ is a prime number. Moreover, p, ${\displaystyle \frac{\left|F\right|}{p}}$ and ${\displaystyle \frac{\left|G\right|}{p}}$ are coprime to each other.

Then, the bismash product ${\mathbb{C}}^G\#\mathbb{C}F$ becomes the tensor product ${\mathbb{C}}^G\otimes \mathbb{C}F$ of Hopf algebras, if

$${\nu }_m({\mathbb{C}}^G\#\mathbb{C}F)={\nu }_m\left(F\right){\nu }_m\left(G\right)(\forall m\ge 0).$$

Proof. 

This is because ${\mathbb{C}}^G\#\mathbb{C}F={\mathbb{C}}^G\otimes \mathbb{C}F$ if and only if ▹ and ◃ are both trivial. □

4. Conclusions

In this paper, we have described the direct and semi-direct products of groups as the particular cases of factorizable groups formulated from a matched pair introduced by Takeuchi in 1981. On the other hand, we have provided some elementary results on higher Frobenius–Schur indicators for the regular representation of a finite group, and related it with the orders of elements.

As the main result, we have found some sufficient conditions when a finite factorizable group is the direct product, if its regular indicators coincide with the product of the regular indicators of its factors. Specifically, the conditions are either that (1) the factor groups have relatively prime orders, (2) or the factor groups are nilpotent, and the greatest common divisor of their orders is a prime appearing once in both orders. Furthermore, some counter-examples have been given when the conditions above are not satisfied.

Moreover, a comparison on the indicators of bismash product Hopf algebras and the corresponding matched pair of groups has been given as an application. In fact, bismash products are generally realized from split abelian extensions of semisimple complex Hopf algebras. As a consequence, our result has provided some conditions when such an extension becomes trivial (meaning that the bismash product is indeed the tensor product), with the help of equalities on values of indicators as a common gauge invariant for Hopf algebras. In other words, this paper can motivate us to determine in a way the structures of particular Hopf algebras under some assumptions on a kind of gauge invariants. Thus, it plays a role in the classification of Hopf algebras satisfying certain properties.