An Organic Puzzle

Here is a puzzle that my ex-brother-in-law, Dodik, gave to me today:

Prove that every group with more than two elements has a non-trivial automorphism.

I usually love puzzles that are solved with a counter-intuitive brilliant idea. This puzzle is different — I didn’t solve it in one elegant swoop. But I still love the puzzle: it feels so natural, and it’s solution feels so natural, that I even decided to call this puzzle “organic.” Or, maybe, I am just in an organic mood today waiting for my organic bananas to be delivered from Boston Organics.

3 Comments

  1. Misha:

    Simple! If all the internal automorphisms are trivial, the group must be commutative, and every commutative group of order >2 has it.

  2. Misha:

    Here is a bit more complicated puzzle: why are fractals everywhere?

  3. Tomasz Wegrzanowski:

    For every element x, y -> x * y * x^-1 is an automorphism. If it’s non-trivial we’re done.

    If it’s trivial, then for all x, y we have x * y * x^-1 = y, or xy=yx, so group is abelian. For abelian groups x -> x^-1 is an automorphism. If it’s nontrivial, we’re done again.

    Otherwise we have: x=x^-1, xy=yx. As this group is abelian, we know it’s isomorphic with a product of group of subgroup generated by {x,y} and corresponding quotient group (all subgroups of an abelian group are normal, so the quotient group must exist).

    Now {e->e, x->y, y->x, xy->xy} is a nontrivial automorphism of the subgroup, and product of it and identity over the quotient group is a nontrivial automorphism of the entire group.

    There’s probably a nicer way that doesn’t require quotient groups anywhere, but I cannot think of it.