Set of Endomorphisms of Non-Abelian Group is not Ring

Theorem

Let $\struct {G, \oplus}$ be a group which is non-abelian.

Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$.

Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as:

$\forall u, v \in \mathbb G: u * v = u \circ v$

where $u \circ v$ is defined as composition of mappings.

Then the algebraic structure $\struct {\mathbb G, \oplus, *}$ is not a ring.

In order to be a ring, it is necessary that the additive operation $\oplus$ is commutative.

However, as $\struct {G, \oplus}$ is specifically defined as being non-abelian, a fortiori $\oplus$ is not commutative.

Hence the result.

$\blacksquare$

1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $10$