# a group

For $G=\langle a,b\ |\ a^2=e,\ b^2=e\rangle$, let us write some elements:

word length : words

0 : $e$

1 : $a,b$

2 : $ab,\quad ba$

3 : $aba,\quad bab$

4 : $abab,\quad baba$

5 : $ababa,\quad babab$

6 : $ababab,\quad bababa$

Note that for odd-length-words, like $aba$ we have: $(aba)(aba)=e$,

but for even-length-words, like $abab$, we have: $(abab)(baba)=e$.

So, this group has infinite many elements of order two and the even-length words form a subgroup, $H$, isomorphic to the group of the integers, and its index is $[G:H]=2$.