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.

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