For any finite group we can define the parity of every element as the parity of the permutation it corresponds to when we view it by the group action of left multiplication on the group set.
But (as far as I can tell) this defintion falls apart for infinite order groups since any permutation of an infinite set is a product of an infinite number of transpositions, which is neither even nor odd (to be clear, I'm defining an "infinite permutation" as just a bijection from an infinite set onto itself)
However I've been reading Coxeter's "Introduction to Geometry", and have learnt that for the group of plane isometries, there is the notion of "sense" or "orientation" by which we can categorise isometries as "direct" or "opposite".
To me this feels like there are far too many similarities to the parity of a group element for them both not to be part of some larger abstraction:
- They both behave the same way under products (direct times direct is direct : even times even is even, etc) - particularly that both the even permutations and the direct isometries form a subgroup.
- Their classifications are both based on expressing the group element as a product of a specific kind of involution (for finite permutations it's transpositions, for isometries it's reflections)
- To build on the previous point - there seems to be a clear connection to the number 2, which I see no reason for (even in the case of plane isometries, the ideas extend to n dimension Euclidean space, so again I see no reason why 2 should be of any relevance)
So I guess my questions are:
What's the bigger pattern that these two phenomena fall under?
Why is 2 relevant to finite groups of any size and euclidean space of any dimension (the latter is the most puzzling to me, since I just don't see why intuitively reflections should force us to leave "rigid motions" and get into a situation where you can't continuously map a shape to its reflection without somehow moving through the next dimension in the transition)