Exercises for Chapter 2

1. Let $a,b \in G$, $G$ a group. Suppose $o(a) = o(b) = o(ab) = 2$. Then show that $ab = ba$.

2. Let $G$ be a group and $H\subset G$, $H \not= \emptyset$. Prove $H \leq G$ if and only if $a,b \in H$ implies $ab^{-1}\in H$. (1 step subgroup test)

3. Let $G$ be a group and let $a \in G$. Let $C_G(a) = \{x \in G\ \vert\ ax = xa\}$. Prove: $C_G(a) \leq G$. This subgroup is called the centralizer of $a$ in $G$.

4. Suppose $G$ is a group which has only one element $a \in G$ such that $o(a) = 2$. Prove that $ax = xa$, for all $x\in G$.

5. (Finite Subgroup Test) Let $H$ be a nonempty finite subset of a group $G$ such that $a,b \in H$ implies $ab \in H$. Then show that $H$ is a subgroup of $G$.

6. Show that the intersection of any collection of subgroups of a group $G$ is a subgroup.

7. Let $G$ be a group. Referring to exercises 3 and 6, the subgroup $Z(G)=\cap_{a\in G}C_G(a)$ is called the center of $G$. Describe in words what $Z(G)$ is, i.e., without using intersections.

8. Show that if $G$ is a finite group, its multiplication table is a Latin square, i.e., each element of the group appears once and only once in each row and in each column of the table.