00:01
So this question did not render well, so i actually went back to cayley's original paper and looked up the three groups so we can write down what they are.
00:13
Three groups of order six.
00:15
So the first one is this one, has a to the sixth equals one, so we have five, six powers of alpha, so alpha is our generator.
00:30
And the second one, we're going to take two generators, alpha and beta, we know that alpha squared is one, beta cubed is one, and alpha and beta commute.
00:44
So alpha beta equals beta alpha.
00:47
And the third one is this one, again alpha squared is one, beta cubed is one, but we have alpha beta is equal to beta squared alpha, so it's actually non -abelian.
01:03
We expect a non -abelian group of order six, it would be s3, the permutations on three elements.
01:12
So anyway, the idea is, so cayley was wrong, and two of these groups are actually isomorphic...