一类pq^2阶群的自同构群

The Automorphism Groups of Certain Groups of Order pq^2

得到了如下定理:设p,q是奇素数,且q<p,则pq2阶群G的自同构群有以下5种情形:(1)｜Aut(G1)｜=q(q-1)(p-1),且Aut(G1) Cp-1×Cq(q-1);(2)｜Aut(G2)｜=(q2-1)(q2-q)(p-1),且Aut(G2) GL(2,q)×Cp-1;(3)｜Aut(G3)｜=pq(p-1),且Aut(G3) Cq×Hol(Cp);(4)｜Aut(G4)｜=pq(p-1)(q-1),且Aut(G4) Hol(Cp)×Hol(Cq);(5)｜Aut(G5)｜=p(p-1),且Aut(G5) Hol(Cp).

The precise structure of the automorphism groups of groups of order pq2 is described, where p and q are odd primes and q<p. That is, Aut(G) satisfies one of the following conditions: (1) |Aut(G1)|=q(q-1)(p-1), Aut(G1)Cp-1×Cq(q-1); (2) |Aut(G2)|=(q2-1)(q2-q)(p-1), Aut(G2)GL(2,q)×Cp-1; (3) |Aut(G3)|=pq(p-1), Aut(G3)Cq×Hol(Cp); (4) |Aut(G4)|=pq(p-1)(q-1), Aut(G4)Hol(Cp)×Hol(Cq); (5) |Aut(G5)|=p(p-1), Aut(G5)Hol(Cp).