QUESTION 3 Is H ={1,2,4} a subgroup of U(7)? Give a reason for you answer. LE10 (Moc)
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- 22. If and are both normal subgroups of , prove that is a normal subgroup of .12. Find all normal subgroups of the quaternion group.Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.
- 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.18. If is a subgroup of , and is a normal subgroup of , prove that .Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.
- (2) Consider S4 and its subgroups H = {i,(12)(34),(13)(24),(14)(23)} and K ={i,(123),(132)}. For a = (1234) ∈ S4 find (H, K) and (K, H) double cosetsWhich of the following cannot be an order of a subgroup of Z12? 12, 3, 0, 4?If N and M are normal subgroup of G,then NM is also a normal subgroup of G.