5. Let G be a group and let a € G. An element b E G is called a conjugate of a if there exists an element x E G such that b = xax-1. Show that any conjugate of a has the same order as a.

Let G be a group and a € G be such that o(r) = 20. Find the orders of a, at, 10, r14 and 30

3. Let G be a group, HAG, KAG, Hn K = {e}. Show that G G/H x G/K.

he following is a multiplication table for a group G. The product a * b is given in the ble where a is from the leftmost column and b from the top row. Let H e V W X Y 2 = e e V W X W V Y Z X W X Y e W X Z V SOSN e What are the orders of the elements? X 3 Y 2 Y 2 Z X V Y e e X W V W e V Y 3 NS Y ১৯৯ {e, z). Show that this is a subgroup of G. Calculate the right cosets of H. Without calculating the left cosets of H, explain why H is a normal subgroup of G. Construct a multiplication table for the quotient group G/H. Is G/H cyclic? Give reasons. Describe the subgroup Z ≤ G/H generated by the element Hx = G/H. What is the order of (G/H)/Z?

Formulate a Cayleyan (multiplication) table and state the reason if why it is a group or not.

Determine whether U(12) with multiplication mod 12 is a cyclic group or not.

plz 2Q and with details.

Define * on Q* by a * b = - ab Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba. 1.3 Let G be a group and Z(G) = {a € G:ax = xa for all x e G}. Show that Z(G)is a normal subgroup of G. Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that o is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)-1 = B-'a-1 . 1.4 1.5

1. Let G and H be groups, and let a: G H be a homomorphism. Show that if a is bijective, then a¹: HG is also a homomorphism.