Groups: Elementary Properties

Some elementary properties of groups are a direct consequence of the group axioms.

Lemma 1

Let $(G, *)$ be a group. Then:

1. The identity element $e_{G}$ of $(G, *)$ is unique.

2. The inverse $a^{-1}$ of $a$ is unique for every $a \in G$.

3. The cancellation laws hold, i.e., for every $a, b, c \in G:$

$$a * b=a * c \Rightarrow b=c \quad \text { and } \quad b * a=c * a \Rightarrow b=c.$$

Proof:

1. Suppose that there are two different identity elements $e_{1}$ and $e_{2} .$ It holds that $e_{1} * e_{2}=e_{1}$ and $e_{1} * e_{2}=e_{2}$ as well. Consequently, it follows that $e_{1}=e_{1} * e_{2}=e_{2}$.

2. Suppose that there are two different inverses $\bar{a}, \hat{a}$ for $a$. Then,

   $$\bar{a}=\bar{a} * e=\bar{a} *(a * \hat{a})=(\bar{a} * a) * \hat{a}=e * \hat{a}=\hat{a}.$$

3. Let $a * b=a * c .$ According to Definition: the group axioms :The_Group_Axioms , there exists an inverse $a^{-1}$ for $a$ such that $a^{-1} * a=e .$ Then, it follows that

   $$b =e * b=(a^{-1} * a) * b=a^{-1} *(a * b)=a^{-1} *(a * c) =(a^{-1} * a) * c=e * c=c.$$

The second statement can be proven in a similar way.

Corollary 1

Let $a \in (G, *).$ If $a^{2}=a$, then $a=e.$

Group homomorphisms

Definition:

Let $G$ and $H$ be groups with their specific binary operations * and $\star$, i.e., $(G, *)$ and $(H, \star) .$ Then, a map $\phi:$ $G \rightarrow H$ is called a group homomorphism if:

$$\phi(a * b)=\phi(a) \star \phi(b)$$

for every $a, b \in G$.

Example:

1. The function $f:(\mathbb{Z},+) \rightarrow(\mathbb{Z},+), x \mapsto 2 x$ is a group homomorphism since $f(x+y)=2(x+y)=2 x+2 y=f(x)+f(y)$ for every $x, y \in \mathbb{Z}$.