Question

If a is the only element of order 2 in a group G, prove that $\text{a}\in \text{Z}\left(\text{G}\right)$.

Short answer

Answer

It is proved that the$\text{a}\in \text{Z}\left(\text{G}\right)$ .

Step-by-step solution

Step-by-Step Solution Step 1: Given information

It is given that a is the only element of order 2 in group G. The center of group Gis denoted by the term Z(G). It is defined as the set of elements of G, which commute with every other element of G.

Prove that a∈ZG

Let $a\in G$be the element of order 2 and b is an arbitrary but fixed element from G.

Use the fact that a²=e. It is known that:

$\begin{array}{c}{\left(b^{-1}ab\right)}^2=b^{-1}{a}^{2}b\\ =b^{-1}b\\ =e\end{array}$

Order of bab^-1 is 2, which is a contradiction because Group G has only one element of order 2. Therefore, .

Now, multiply both the sides in bab^-1 on the right side by b.

$\begin{array}{c}ba{b}^{-1}b=ab\\ ba=ab\left(\text{since\hspace{0.17em}}b{b}^{-1}=e\right)\end{array}$

This shows that element a commutes with every element of group G.

Thus, it is proved that $a\in Z\left(G\right)$.