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Sunday, November 10, 2013

Automorphism Computation (Research)

MathJax TeX Test Page Compute |Aut(Z42×D8)|.

Let A=Z42, B=D8, and G=Aut(A×B). We note A is abelian, so we consider F=AutA(A×B)G. We also note Z(A×B)=Z42×r2Z52. Since A×1Z(A×B) char A×B we observe [G : F]=[AutA×B(Z(A×B)) : AutA×BA(Z(A×B))].

Let ΦG. Then we note Φ(r2)=Φ(r)2 so since r2 and also Φ(r2) are in Z(A×B) we have Φ(r2)=r2 (r2 is the only nonidentity square in Z(A×B)). Conversely, let σAut(Z(A×B)) be such that σ(r2)=r2. We observe the relations of A×B generated in a,b,c,d,s,r:a2=b2=c2=d2=s2=r4=1aba1b1=aca1c1=ada1d1=asa1s1=ara1r1=...=1(rs)2=1where the second line lists all the commutation relations (all commute but r, s). Some extending map σ defined by its action on a,...,r,s is an automorphism iff σ(a), ... , σ(r),σ(s) generate A×B and σ is one on these relations generators in the free group. If we let σ act on a, b, c, d as σ does and set σ(s)=s and σ(r)=r we observe σ(r2)=σ(r2)=r2 and it follows that σ extends σ, and is an automorphism since the orders line is fulfilled (Z(A×B) has exponent 2, so σ(a)2=σ(a)2=1 and similar for b,c,d and clearly σ(s)2=σ(r)4=1), the commutation line is fulfilled (all of these relations involve a, b, c, or d, and σ preserves their centricity), the last relation is fulfilled (σ fixes D8), and the generation is established as a,...,d is in the image of σ on Z(A×B), hence σ on A×B, plus s=σ(s) and r=σ(r). Thus AutA×B(Z(A×B)) is isomorphic to the group of isomorphisms fixing the last coordinate of Z52, whose order we compute (2521)(2522)(2523)(2524)=322,560.

Utilizing the previous reasoning we note that σAutA(Z(A×B)) extends iff σ(r2)=r2, so |AutA×BA(Z(A×B))| is the number of automorphisms of Z52 fixing the last coordinate and restricting to an automorphism of Z42×1, i.e. the number of automorphims of Z42 which is computed to be (2420)(2421)(2422)(2423)=20,160. Thus we calculate [AutA×B(Z(A×B)) : AutA×BA(Z(A×B))]=16 distinct cosets of AutA(A×B) in Aut(A×B).

We have seen |AutA(A×B)|=|Aut(A)|·|Hom(B,A)|·|Aut(B)|. We have already computed |Aut(Z42)|=20,160. As well, utilizing the relations for D8 yields |Aut(D8)|=8. We see Hom(B,A) is the set of all maps defined freely on r and s in the free group factoring through the relations of D8, i.e. mapping one on the relations of D8, and since Z42 has exponent 2 we see every such mapping is a homomorphism, so |Hom(B,A)|=162=256. |Aut(A×B)|=|G|=[G : F]·|F|=[G : F]·|Aut(A)|·|Hom(B,A)|·|Aut(B)|=16·20,160·256·8=660,602,880
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Alternatively, we calculate the number of automorphisms of the characteristic center that extend |AutA×B(Z42×r2)|=322,560 (see above) which represent the cosets of the subgroup of automorphisms that fix the center. We see that sending r to either r or r3 with any coordinates for a, b, c, d together with any mapping of s to s, rs, r2s, or r3s and any coordinates of a, b, c, d are all automorphisms that fix the center, so 2,048·322,560=660,602,880.

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