Stability of Maximum Functional Equation and Some Properties of Groups

: In this research paper, we deal with the problem of determining the function χ : G → R , which is the solution to the maximum functional equation (MFE) max { χ ( xy ) , χ ( xy − 1 ) } = χ ( x ) χ ( y ) , when the domain is a discretely normed abelian group or any arbitrary group G . We also analyse the stability of the maximum functional equation max { χ ( xy ) , χ ( xy − 1 ) } = χ ( x ) + χ ( y ) and its solutions for the function χ : G → R , where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G .

We recommend the readers to see [3,4], and the references cited therein in order to obtain comprehensive results related to functional Equation (1), which characterises the additive function's absolute value.
This paper is arranged, as follows: in Section 2, we prove the functional Equation (1) for any arbitrary group G without any characterisation of an additive function's absolute value. Besides, we also demonstrate some consistent and useful results concerning the normal subgroup of G.
In Section 3, we analyse the functional Equation (2) in order to obtain its solution. For this purpose, we drop the additional assumption that G is an additive abelian group and is divisible by 6. We present the generalization of Theorem 2 by proposing a discretely normed abelian group G. Moreover, we investigate the functional Equation (2) for any arbitrary group G, which satisfies the Kannappan condition [10]. Section 4 deals with the stability results of the MFE (1) for a function χ : G → R, where G be any group and find some useful connection of stability with commutators and embedding of a group. (1) Throughout this article, let G be any group, and 1 is considered to be the identity element of a group G. Definition 1 ([10]). Let G be an arbitrary group, we say a function χ : G → R satisfies the Kannappan condition if

Solutions of the Functional Equation
for all u, v, x ∈ G.
Theorem 3. Let G be a group and function χ : G → R satisfies the Kannappan condition then if and only if and also satisfies for any x ∈ G.

Corollary 2.
Let G be a group and function χ : G → R satisfies the Kannappan condition, and then χ is a solution of equation for any x, y ∈ G and also satisfies for any x ∈ G if and only if there exists an additive function α : G → R such that χ(x) = |α(x)| for any x ∈ G.
Proof. Because the function χ : G → R satisfies the Equations (4) and (5), then by Theorem 3 for any x, y ∈ G.
Proof. Let x ∈ G be an arbitrary and assume that v ∈ N χ . We can deduce from (3) for any x ∈ G and v ∈ N χ . Moreover, we already proved that χ(x −1 vx) = χ(v) for any x ∈ G and v ∈ N χ , then, by replacing v with xv, we can evaluate that χ(vx) = χ(vx) for any x ∈ G and v ∈ N χ . Additionally, N χ is a normal subgroup of G; therefore, G/N χ is an abelian quotient group.

Solutions of the Functional Equation
if and only if simultaneously holds and satisfies Proof. Suppose that χ satisfies Equation (9). Assume 1 as neutral element of group G, then by putting x = y = 1 in (9), we can obtain that χ(1)χ(1) = χ(1), then χ(1) = 0 or χ(1) = 1. Let χ(1) = 0, then by Equation (9) we can compute χ(x) = 0 for any x ∈ G. Assume that χ(1) = 1, then from (9), It is easy to see that 2 for any x ∈ G. The proof of Equation (10) consists of the following simple computation: Conversely, suppose that the Equations (10) and (11) are satisfied and χ(x) = 0. Subsequently, it can be determined that Here, we have two cases, in the first case, using Equation (10) and (11), we derive the required result, as follows: From second case, when χ(x) < χ(y), then we can also get that max{ χ(xy), Corollary 5. Let G be a group and function χ : G → R satisfies the maximum Equation (9) if and only if and also χ(1) = 1.

Corollary 6.
Let G be a group and function χ : G → R is a solution of Equation (10) satisfies (11) if and only if there exists an additive function α : G → R, such that χ(x) = e |α(x)| for any x ∈ G.

Definition 2 ([11]
). Let G be an abelian group. Subsequently, a function χ : G → R is said to be a discrete norm on G if there exists some α > 0 such that χ(x) > α whenever x is not identity element of G. Afterwards, (G, χ) is called the discretely normed abelian group [11].
Simon and Volkmann [5] proved Theorem 2 with additional assumption that G is an additive abelian group and is divisible by 6, but we present the generalization of Theorem 2 by introducing the notion of discretely normed abelian group G, as follows: Theorem 6. Assume that (G, χ) be a discretely normed abelian group, then a function χ : G → R fulfills (9) if and only if there exists an additive function α : G → R, such that χ(x) = e |α(x)| for any x ∈ G \ {1}.
Proof. Because G is a discretely normed abelian group, therefore there exists a discrete norm function  Proof. From Theorem 6, we concluded that maximum functional Equation (9) holds; therefore, using Theorem 5, we can also obtain required proof.
We have well-known theorem presented by Steprāns Juris in [11] about a group G, which is a discretely normed abelian group. Therefore, we have following corollaries.  Theorem 7. Let G be any group and let a function χ : G → R is a solution of Equation (10) and (11), which is not identically zero, then there exists a normal subgroup H Proof. Because the function χ : G → R satisfying the Equations (10) and (11), then by Theorem 5, we can obtain that for all x, y ∈ G.

Corollary 10.
For any group G, let a function χ on group G satisfying (13), which is not identically zero, then χ : G/H χ → R also satisfies for any x ∈ G, v ∈ H χ . Moreover, Proof. Let x ∈ G be an arbitrary and assume that v ∈ H χ , then χ(v 2 ) = 1. We can deduce from (13) that Because H χ is a subgroup of G, then v 2 ∈ H χ , so we can see that χ(xv 2 ) = χ(x)χ(v 2 ) for any x ∈ G and v 2 ∈ H χ , therefore χ(xv 2 ) = χ(x). Writing x instead of y and v instead of x in (13), we can conclude Corollary 11. Let G be a group and function χ : G → R satisfies the Kannappan condition. If the maximum functional Equation (13) holds, then min{ χ(xy), χ(xy −1 ) } ∈ H χ for any x, y ∈ G.

Stability of Maximum Functional Equation
for some λ ≥ 0. Subsequently, we can obtain a solution χ : Additionally, χ is given by Additionally, by (18), χ is uniquely determined, and by (19), the requirement of χ − η to be bounded is also satisfied.
Writing x 2 instead of x, then also −λ ≤ η(x 2 ), so we can get that From (17), we have Combining both cases, we can conclude From (17), we have max{η(x 2 ), η(1)} ≤ λ + 2η(x), which is only possible when From inequalities (23) and (24), we have When we observe (25), it can be seen that the function χ : G → R presented in (20) exists and this χ satisfies also χ fulfills (19). Furthermore, writing x 2 n instead of x in (21), dividing by 2 n and taking a limit n → ∞ and, using (20), we can obtain that χ(x) ≥ 0 for any x ∈ G. Hence we can obtain (18) from (26). Furthermore, when we consider (18), then we can easily see the uniqueness of χ, due to the result that By utilizing Theorem (8), we are going to derive the stability of Equation (1) in two-variables x and y. Theorem 9. Let G be any group and function η : for some λ ≥ 0. Subsequently, we can evaluate a unique solution χ : G → R of max{χ(xy), χ(xy −1 )} = χ(x) + χ(y) for any x, y ∈ G (28) Additionally, Proof. Applying the Theorem (8), we can easily prove the required results. First, writing x instead of y in (27), then, from Theorem (8), we can obtain a function χ : G → R. We also need to prove that this function χ satisfies (28). To obtain the required maximum functional Equation (28), we need to write x 2 n instead of x and y 2 n instead of y in (27), dividing inequality by 2 n and, applying limit n → ∞ and also using Equation (30).
Theorem 10. Let G be a group and function η : G → R fulfills (27), then, for any x, y ∈ G, it holds the condition if and only if the Equation (28) is satisfied.

Remark 1.
(i) The given condition (31) holds when group G is an n-Abelian group (for any integer n, a group G is called an n-Abelian group if (ab) n = a n b n , for any a, b ∈ G, see [12,13] ).
(ii) The condition (31) also satisfies when group G belongs to the class C n for any natural number n ∈ N (for all n ∈ N, C n is denoted as the class of groups, which satisfies the relation b n a n = a n b n ).
(iii) When χ is central, then condition (31) also holds. (1) is stable on group G, then every free abelian group K can be embedded into a group G.

Corollary 13. If maximum functional Equation
Proof. Because K is a discretely normed abelian group, then, by applying theorem from [11], K is a free group; hence it can be embedded into a group G.

Theorem 12.
A function η : G → R satisfying (27) is bounded on the commutator group G 1 of the subgroup G of G if the condition (31) is satisfied.
Proof. Assume that condition (31) is satisfied. Subsequently, by Theorem 10, we can get that χ(x) + χ(y) = max{χ(xy), χ(xy −1 )} holds for any x, y ∈ G. Because this maximum equation holds; therefore, by Theorem 1, there exists an additive function α : G → R, such that χ(x) = |α(x)| for any x ∈ G. For a, b ∈ G, take a −1 b −1 ab ∈ G, then Because χ is zero on the commutator group G 1 of the subgroup G of G, η is bounded on commutator group G 1 .