exercise 1.1

todo

**11. The following conditions on a group $G$ are equivalent:

• (i) $G$ is abelian;
• (ii) $(ab{)}^2$ $=a^2{b}^2$ for all $a,b\in G$;
• (iii) $(ab{)}^{-1}=a^{-1}{b}^{-1}$ for all $a,b\in G$;
• (iv) ${\left(ab\right)}^n=a^n{b}^n$ for all $n\in \mathbf{Z}$ and all $a,b\in G$;
• (v) $(ab{)}^n=a^n{b}^n$ for three consecutive integers $n$ and all $a,b\in G$.

Show that (v) $\Rightarrow$ (i) is false if “three” is replaced by “two.”

13. If $a^2=e$ for all elements $a$ of a group $G$, then $G$ is abelian.

\begin{proof} For any $a,b\in G$, we have $[a,b]=1$. \end{proof}

14. If $G$ is a finite group of even order, then $G$ contains an element $a\ne e$ such that $a^2=e$.

\begin{proof} See here. \end{proof}

15. Let $G$ be a nonempty finite set with an associative binary operation such that for all $a,b,c\in G$, $ab=ac\Rightarrow b=c$ and $ba=ca\Rightarrow b=c$. Then $G$ is a group. Show that this conclusion may be false if $G$ is infinite.

16. Let $a_1,a_2,\dots$ be a sequence of elements in a semigroup $G$. Then there exists a unique function $\psi :{\mathbf{N}}^{\ast }\to G$ such that $\psi (1)=a_1,\psi (2)=a_1{a}_2,\psi (3)=\left(a_1{a}_2\right)a_3$ and for $n\ge 1,\psi (n+1)=(\psi (n))a_{n+1}$. Note that $\psi (n)$ is precisely the standard $n$ product ${\prod }_{i=1}^n{a}_i$. (Hint: Applying the Recursion Theorem 6.2 of the Introduction with $a=a_1,S=G$ and $f_n:G\to G$ given by $x\mapsto x{a}_{n+2}$ yields a function $\phi :\mathbf{N}\to G$. Let $\psi =\phi \theta$, where $\theta :{\mathbf{N}}^{\ast }\to \mathbf{N}$ is given by $k\mapsto k-1$.)