If |G|=55, and G is non- abelian then how many subgroups of order 5 None of them O 10 55 O 60 11 O 50 LO

1. Which of the following is a nontrivial subgroup of Z3 x Z3 where Z3 is the additive group of integers modulo 3 consisting of {0,1,2} and the Cartesian product is the set of ordered pairs?   a) {(0,0),(2,2)}   b) {(0,0),(1,2),(2,1)}   c) {(0,0),(1,1),(1,2)}   d) {0,1} 2. Which of the following is a nontrivial subgroup of the quaternion group of order 8? This group is plus or minus 1, J, K, L where signs multiply as usual, 1 is a multiplicative identity, the squares of J, K, L are -1 and JK=L, KJ=-L, KL=J, LK=-J, LJ=K, JL=-K. The subgroup in question is cyclic.   a) {J,-K,L}   b) {1,-J,K,L}   c) {1,L}   d) {1,K,-1,-K} 3. For the following two functions, which of these 4 alternatives is correct? f(x)=x3 -x from the real numbers to itself and g(x)={(1,1),(2,3),(3,1)} from the set {1,2,3} to itself.   a) both are one-to-one   b) the first, but not the second, is…

7 Suppose that a E G. For each y in the conjugacy class of a, let G(a y)= {g E G: gag= y} be the set of group elements which conjugate a into y. Prove that G(a y) is a coset of CG(a) 1

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