Hello, I have an abstract algebra question.

I need your help.

Thank you so much.

*2. (6 points) Let N and H be subgroups of a group C. By Q1(a) on Problem Sheet 1, N 0 H is a subgroup of G, and hence a subgroup of both N and H. (a) Show that N/(NDH) ) G/H, a(NH) r> aH is a well-dened injection. In particular, we have (N : N F] H) S (G : H). (b) Suppose now that (G : N) = 2, and that H has nite index in G. Show that (N: N H H) is equal to (G' : H) if H 5; N, and equal to (G' : H)/2 otherwise (Hint: Use (a) and Q1). *7. (4 points) Let G be a group, and let x E G. Recall that we write Zo(x) for the centralizer of x in G, i.e., the subgroup of G consisting of all elements that commute with x (see Problem Sheet 1, Q3). Show that if C(x) is the conjugacy class of x in G, then the map f : G/ZG(x) + C(x), aZG(x) axa-1 is a well-defined bijection. In particular, we have |C(x) | = (G : ZG(x)).*8. (8 points) Let n be an integer 2 3, and let p E An. Let C denote the conjugacy class of p in Sn, and C' Q C the conjugacy class of p in An. We saw in class that C is the subset of 8,, consisting of all permutations with the same cycle type as p. In An, the situation is more complicated, i.e., C' need not equal C in general. (a) Show that C' 7g C if and only if ZS" (p) g An, and that |C'| = |C|/2 when this holds (Hint: Q7 and Q2). (b) Deduce that C' = C in each of the following cases: (i) p xes more than one element of {1, . . . , 11}; (ii) The cycle type of p contains an even integer; (iii) The cycle type of p contains at least one repeated integer. (C) Deduce that if n 2 5, then the 3cycles in Sn form a conjugacy class in An. (Remark: The conditions in (b) are not only sufcient, but necessary, i.e., C' = C if and only if one of these conditions is satised. Feel free to prove this stronger statement if you can)