5.5. TRANSITIVE SUBGROUPS OF S 5 259 (a) Show that ker .±/ is generated by an element of Z 2 ± Z 2 of order 2 . (b) Show that ± is determined by its kernel. That is, if ± and ±0 are two such homomorphisms with the same kernel, then ± D ±0 . (c) Show that if ˇ and ± are two such homomorphisms, then there exists an automorphism ' 2 Aut . Z 2 ± Z 2 / such that ker .ˇ/ D ker .± ı '/ . Consequently, ˇ D ± ı ' . (d) Conclude from Exercise 5.4.13 that if ˇ and ± are two non-trivial homomorphisms from Z 2 ± Z 2 into Aut . Z 7 / , then Z 7 Ì ˇ . Z 2 ± Z 2 / Š Z 7 Ì ± . Z 2 ± Z 2 /: 5.4.15. Classify the non-abelian group(s) of order 20. 5.4.16. Classify the non-abelian group(s) of order 18. 5.4.17. Classify the non-abelian group(s) of order 12. 5.4.18. Let p be the largest prime dividing the order of a ﬁnite group G , and let P be a p –Sylow subgroup of G . Find an example showing that P need not be normal in G . 5.5. Application: Transitive Subgroups of S 5 This section can be omitted without loss of continuity. However, in the
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