Let G = Sn. Assume that H is a subgroup of G, that H contains atleast one odd permutation, and that |H| > 2. Show that H has somenon-trivial normal subgroup.
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Let G = Sn. Assume that H is a subgroup of G, that H contains at
least one odd permutation, and that |H| > 2. Show that H has some
non-trivial normal subgroup.
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- 19. With and as in Exercise 18, prove that is a subgroup of . Exercise18: 18. If is a subgroup of , and is a normal subgroup of , prove that .With H and K as in Exercise 18, prove that K is a normal subgroup of HK. Exercise18: If H is a subgroup of G, and K is a normal subgroup of G, prove that HK=KH.22. If and are both normal subgroups of , prove that is a normal subgroup of .
- 27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .16. Let be a subgroup of and assume that every left coset of in is equal to a right coset of in . Prove that is a normal subgroup of .Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.