Give an example of an abelian group of order 4 in which ever

Give an example of an abelian group of order 4 in which every nonidentity element a satisfies a*a=e.

Solution

we can write this group another way:

V = <a.b> a^2 = b^2 = e; ab = ba.

the only elements V has is {e,a,b,ab} because:

(ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = a^2b^2 = ee = e.

still another way to write this group is the 4 matrices:

[1 0]..[-1 0]..[-1 0]..[ 1 0]
[0 1], [0 -1], [0 1 ], [0 -1]

or as the following set of 4 bijections on the set X = {a,b,c,d}

a<-->a
b<-->b
c<-->c
d<-->d (the identity map)

a<-->b
b<-->a
c<-->d
d<-->c (swap a and b, and swap c and d)

a<-->c
b<-->d
c<-->a
d<-->b (swap a and c, swap b and d)

a<-->d
b<-->c
c<-->b
d<-->a (swap and d, swap b and c).

(try this with four objects, see that every \"pair of pair-swaps\" followed by another one, either gives you the remaining pair-swap, or your original configuration).