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**Deveno** sometimes the extreme abstraction of group theory overwhelms people. it involves a bit of "un-learning", because operations like + or * no longer act in the ways we have been drilled into accepting as "self-evident".

the integers modulo n are somewhat helpful, because they are a concrete representation of cyclic groups, and ordinary things we learned from 3rd grade (or so) are still helpful in guiding our intuition (ok, 3 divides 6, so i believe that {0,3} is a subgroup of {0,1,2,3,4,5}).

although cyclic groups are important (they are the "prime numbers" of the abelian groups), they are still rather specialized. abelian groups are also rather specialized, they are "almost" vector spaces (the technical term here is Z-module). non-abelian groups are, in some sense, the most "typical" (the worst case).

the smallest non-abelian group is S3, which has only a scant 6 elements. it is a worth-while exercise to do the following:

version 1: imagine you have a set of 3 elements {a,b,c}. describe all possible bijections on this set.

version 2: imagine you have an equilateral triangle in R^2, with vertices at (1,0), (-1/2, √3/2), and (-1/2, -√3/2). describe all 2x2 matrices that fix these 3 points.

version 3: imagine you have the free group on 2 letters (formally, these are all strings such as aabbbab*a*a*ba, etc, with the proviso that aa* = bb* = a*a = b*b = [ ], the empty word, the multiplication is "concatenation"), subject to the following rules for "reduction": aaa = bb = [ ], ba = aab. describe all possible words.

now, convince yourself these are all "the same thing". are they all groups? what are the possible subgroups?

if any subgroups exist, explicitly calculate left and right cosets, xH and Hx. for which subgroup(s) are these two cosets always the same? express this in terms of:

1) how many elements are "fixed" by the elements of the subgroup.

2) the "kind" of geometric operation it is.

3) a's and b's.

these examples were not chosen at random. (1) corresponds to an "action" view, (2) corresponds to a "matrix" view (i think you can see the physical significance of this), and (3) corresponds to an "elements" view. in order of abstraction, we have (2), (1), (3). with any group, all 3 views are possible, and sometimes we can gain unique information by getting a result from one view, and applying it to another.