Question

If every element of a group G is its own inverse, then G is

• A. finite
• B. infinite
• C. not abelian
• D. abelian

Hint:

If every element is its own inverse, the group is abelian (Boolean group).

Answer 1

The Correct Option is D

Step 1: Basic Principle
$a = a^{-1}$ means $a^2 = e$ for all $a \in G$.
Step 2: Solution Procedure:
Take any $a,b \in G$. Then $(ab)^2 = e \Rightarrow abab = e$. Multiply on left by $a$: $a(abab) = a \Rightarrow (a^2)bab = a \Rightarrow bab = a$. Multiply on right by $b$: $(bab)b = ab \Rightarrow ba(b^2) = ab \Rightarrow ba = ab$. So $ab = ba$, thus G is abelian.
Step 3: Required Answer:
G is abelian.