exercise of chapter 1.9

1. Every nonidentity element in a free group $F$ has infinite order.
2. Show that the free group on the set $\{a\}$ is an infinite cyclic group, and hence isomorphic to $\mathbf{Z}$.
3. Let $F$ be a free group and let $N$ be the subgroup generated by the set $\{x^n\mid x\in F$, $n$ a fixed integer $\}$. Show that $N◃F$.
4. Let $F$ be the free group on the set $X$, and let $Y\subset X$. If $H$ is the smallest normal subgroup of $F$ containing $Y$, then $F/H$ is a free group.
5. The group defined by generators $a,b$ and relations $a^8=b^2{a}^4=a{b}^{-1}ab=e$ has order at most 16 .
6. The cyclic group of order 6 is the group defined by generators $a,b$ and relations $a^2=b^3=a^{-1}{b}^{-1}ab=e$.
7. Show that the group defined by generators $a,b$ and relations $a^2=e,b^3=e$ is infinite and nonabelian.
8. The group defined by generators $a,b$ and relations $a^n=e\left(3\le n\epsilon N^{\ast }\right),b^2=e$ and $abab=e$ is the dihedral group $D_n$. [See Theorem 6.13.]
9. The group defined by the generator $b$ and the relation $b^m=e\left(m\in N^{\ast }\right)$ is the cyclic group $Z_m$.
10. The operation of free product is commutative and associative: for any groups $A,B,C,A\ast B\cong B\ast A$ and $A\ast (B\ast C)\cong (A\ast B)\ast C$.
11. If $N$ is the normal subgroup of $A\ast B$ generated by $A$, then $(A\ast B)/N\cong B$.
12. If $G$ and $H$ each have more than one element, then $G\ast H$ is an infinite group with center $⟨e⟩$.
13. A free group is a free product of infinite cyclic groups.
14. If $G$ is the group defined by generators $a,b$ and relations $a^2=e,b^3=e$, then $G\cong Z_2\ast Z_3$. [See Exercise 12 and compare Exercise 6.]
15. If $f:G_1\to G_2$ and $g:H_1\to H_2$ are homomorphisms of groups, then there is a unique homomorphism $h:G_1\ast H_1\to G_2\ast H_2$ such that $h\mid G_1=f$ and $h\mid H_1=g$.