In working on a paper, I found myself in the amusing but unusual situation of having a group , a subgroup and an element such that
This certainly can happen: the two obvious cases are when , or when is an involution that happens to be in the normalizer of .
In fact the general case is just a tweak of this last case: we have if and only if and , or in other words, if belongs to the normalizer, and is an involution modulo .
This is of course easy to check. I then asked myself: what about the possibility that
where and are arbitrary integers? Can one classify when this happens? The answer is another simple exercise (that I will probably use when I teach Algeba I next semester): this is the case if and only if and , where is the gcd of and . In particular, for all pairs where and are coprime, the condition above implies that belongs to the normalizer of .
Here is the brief argument: having fixed , let
This set is easily seen to be a subgroup of . Furthermore, note that implies that , which in turns means that and .
Hence if , we get
contains , which is just
where . But means exactly that .
Thus we have got the first implication. Conversely, the conclusion means exactly that
shows that .
To finish, how did I get to this situation? This can arise quite naturally as follows: one has a collection of representations of a fixed group , and an action of a group latex on these representations (action up to isomorphism really).
For a given representation , we can then define a group
and also a subset
where denotes the contragredient representation.
It can be that is empty, but let us assume it is not. Then has two properties: (1) it is a coset of (because acts on simply transitively); (2) we have for all (because the contragredient of the contragredient is the representation itself).
This means that, for some , we have , and furthermore
since for all .
By the previous discussion, we therefore get a good handle on the structure of : either it is empty, or it is of the form for some such that and normalizes in . In particular, if is trivial (which happens often), either is empty, or it consists of a single element which is an involution of .