Automorphisms of D8

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 >
εtriviallyε
(b a2b) (ab a3b)conjugation by a or by a3s2
(a a3) (b a2b)conjugation by ab or by a3bt
(a a3b) (ab a3b)conjugation by a or by a3s2t
(b ab) (a2b a3b)not innerst
(b a3b) (a2b ab)not inners3t
(a a3) (b ab a2b a3b)not inners
(a a3) (b a3b a2b ab)not inners3

Thus we see that its automorphism group is itself isomorphic to D8, and its inner automorphism group is isomorphic to C2×C2.

Possibilities for D8 ⋊ 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