College Algebra Exam Review 246

College Algebra Exam Review 246 - 256 5 ACTIONS OF GROUPS...

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256 5. ACTIONS OF GROUPS distinct conjugates is trivial (as the size must be a divisor of the prime 5). Therefore, the union of conjugates of R contains 6 4 D 24 elements of order 5. Likewise, if Q is not normal, then the union of its 10 conjugates contains 20 elements of order 3. Since G has only 30 elements, it is not possible for both R and Q to be non-normal. Since at least one of R and Q is normal, N D RQ is a subgroup of G of order 15. Now N is normal in G , since it has index 2, and cyclic, since any group of order 15 is cyclic. We have G D NP and N \ P D f e g , so according to Corollary 3.2.5 , there is a homomorphism ˛ W Z 2 ! Aut . Z 15 / , such that G Š Z 15 Ì ˛ Z 2 . To complete the classification of groups of order 30, we have to classify such homomorphisms; the nontrivial homomorphisms are determined by order 2 elements of Aut . Z 15 / . We have Aut . Z 15 / Š Aut . Z 5 / Aut . Z 3 / Š ˚.5/ ˚.3/ Š Z 4 Z 2 . In particular, if is an automorphism of Z 5 Z 3 , then there exist unique automorphisms 0 of Z 5 and 00 of Z 3 such that for all .OEaŁ; OEbŁ/
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