A2 - elementary properties of a group

uniqueness of identity

in a group $G$ , there is only one identity

suppose $G$ has two identities $e_{1}, e_{2}$ such that $e_{1} \neq e_{2}$

\begin{matrix} a e_{1} = e_{1} a = a \\ ⟹ e_{2} e_{1} = e_{2} \\ a e_{2} = e_{2} a = a \\ ⟹ e_{1} e_{2} = e_{1} \\ ⟹ e_{1} = e_{2} \end{matrix}

this contradicts the original assertion

right and left cancellation

if $a, b, c \in G :$
- right cancellation: $b a = c a ⟹ b = c$
- left cancellation: $a b = a c ⟹ b = c$

since $a \in G$ , there exists $a^{'} \in G$ such that $a a^{'} = e$

\begin{matrix} b a = c a \\ b (a a^{'}) = c (a a^{'}) \\ b e = c e \\ ⟹ b = c \end{matrix}

uniqueness of inverses

if $G$ has an identity $e$ , and $a \in G$ , there exists a unique $b \in G$ such that $a b = b a = e$

let $b_{1}$ and $b_{2} \in G$ such that $b_{1} \neq b_{2}$

\begin{matrix} b_{1} b_{2} = b_{2} b_{1} = e \\ b_{2} b_{1} = b_{1} b_{2} = e \\ ⟹ b_{1} = b_{2} \end{matrix}

inverse of the product

if $a, b \in G$ then $(a b)^{'} = b^{'} a^{'}$

\begin{matrix} LHS a b = (a b)^{'} a b = e \\ RHS a b = b^{'} a^{'} a b = b^{'} e b = b^{'} b = e \end{matrix}