By Flaass D.G.
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Extra info for 2-Local subgroups of Fischer groups
This pair of elements of order 2 therefore generate all 2n rigid symmetries of a regular n sided figure. Thus there are dihedral group of order 2n when n is finite and bigger than 2. The infinite case is very similar; the number line becomes the regular 00gon when we declare a vertex at each integer. 17. Consider the following two rigid symmetries of this object: (a) reflection about 0 (the map sending the vertex at a to -a for every a E Z, and (b) reflection about 1/2 (the map sending the vertex at b to -b+ 1.
Iv) This is a triviality, since x = 1 . x E H x. 18 gives a criterion for deciding when two (right) cosets of H in G are equal. If two right cosets of H in G are not equal, then something very interesting happens. 19 Suppose that H is a subgroup of G, and x, y E G. It follows that either H x = H y or H x n H y = 0. Proof Either HxnHy = 0 or there exists z E HxnHy in which case zx- 1 , zy-l E H. 18(ii) we have Hx = Hy. o Thus a pair of right cosets of H in G are either disjoint or equal. 20 Suppose that H is a subgroup of G.
This is why we have called this group V. It has exactly 5 subgroups; itself, the trivial subgroup, and three subgroups of order 2. The intersection of any pair of distinct subgroups of order 2 is the trivial group, and the join of any pair of distinct subgroups of order 2 is V. 2. Some group theorists find it very convenient to think about groups in terms of these diagrams. In particular, the isomorphism theorems which we will prove in Chapter 2 have very pleasant interpretations in terms of Hasse diagrams.
2-Local subgroups of Fischer groups by Flaass D.G.