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D8 can be presented as < a,b | a4, b2, abab >. It has eight automorphisms, which permute its elements {ε a a2 a3 b ab a2b a3b} as follows:
| permutation of D8 | inner automorphism? | element of automorphism group presented as < s,t | s4, t2, stst > |
|---|---|---|
| ε | trivially | ε |
| (b a2b) (ab a3b) | conjugation by a or by a3 | s2 |
| (a a3) (b a2b) | conjugation by ab or by a3b | t |
| (a a3b) (ab a3b) | conjugation by a or by a3 | s2t |
| (b ab) (a2b a3b) | not inner | st |
| (b a3b) (a2b ab) | not inner | s3t |
| (a a3) (b ab a2b a3b) | not inner | s |
| (a a3) (b a3b a2b ab) | not inner | s3 |
Thus we see that its automorphism group is itself isomorphic to D8, and its inner automorphism group is isomorphic to C2×C2.
To build D8 ⋊ C2 we need a C2 which is a subgroup of aut(D8). Writing aut(D8) in terms of s and t as above, we find five possible subgroups of order 2: {1, s2}, {1, t}, {1 s2t}, {1, st}, {1, s3t}.
If we use {1, s2} we get the Pauli group.
If we use {1, t} or {1 s2t} we get D8×C2.
If we use {1, st} or {1 s3t} we get D16.
We should expect {1, t} and {1 s2t} to be unproductive: these are equivalent to conjugation by ab and b respectively, and so yield nothing interesting when used to extend D8. In effect we have shown that D8⋊C2 ≅ (C4⋊C2)⋊C2 ≅ C4⋊(C2×C2) ≅ (C4⋊C2)×C2 ≅ D8×C2.
It may appear surprising that {1, s2} does produce an interesting result, even though the automorphism corrsesponding to s2 is an internal one, produced by conjugation by a. However, this automorphism has order 2, but is produced by conjugation with an element of order greater than 2, so no sequence of isomorphisms such as the one given above for D8⋊C2 is possible.
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Copyright N.S.Wedd 2008