College Algebra Exam Review 246

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 Z 15 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 / Aut . Z 5 / Aut . Z 3 / .5/ .3/ Z 4 Z 2 ....
View Full Document

Ask a homework question - tutors are online