On Tue, 23 Sep 2003, John Conway wrote:
On Mon, 22 Sep 2003, Dan Hoey wrote:
I found two such groups of order 16, but I haven't figured out what their names are. A presentation for the first is
<a,b : a^2 = b^4 = (b a b)^2 = (b^-1 a b a)^2 = 1> .
Its squares are 1, b^2, and (a b)^2, and the product of the last two is not a square.
I don't immediately recognise this group in that presentation. I'll play with it offline to see which one it is.
I've now done so. As I expected, it's what I think of as "the last group" of order 16, which Coxeter calls (4,4|2,2) (except that I may have the wrong style of bracket or punctuation): 1 = a^4 = b^4 = (ab)^2 = (a^-1.b)^2. [This is more symmetrical than the presentation you give - my a can be taken as your ab.] The elements can be written a^i.b^j, and exactly half of them (those for which i+j is odd) have order 4. They are of two "sexes" according as i is odd and j even or i even and j odd, and any two elements of opposite sexes generate the group in the above presentation. The square of an element of order 4 is a^2 or b^2, depending only on its sex, so of course it's an example. [I remark that a^2 and b^2 generate the center.]
The other group may be presented <a,b : a^4 = b^4 = a b a^-1 b = 1>.
Its squares are 1, a^2, b^2, and again the product of the two non-identity squares is not a square.
This group is called 4:4 in my system. [In general, A:B denotes the split extension of a group of structure A by one of structure B, and it is usually assumed that it's not the direct product, since we'd call that A x B.]
This group is much easier to think about, since it's that split extension. The square of a^i.b^j is a^(2i) if j is even, and b^2 if j is odd, so superficially it looks very like the other (in the new presentation) in respect of this behavior, though this time there's no symmetry between a and b. When I guessed there wouldn't be examples of order 16, I planned to check (4,4|2,2) just in case, because it's the most interesting one, but forgot. [I "learned" the groups of order 16 by heart long ago, though now I may be a bit shaky. The standard reference for them is a table in the back of Coxeter and Moser, but they describe them in ways that don't tell you the structure, so it's better to use ATLAS-type notations that do.] John Conway