Let H denote the smallest subgroup of S(5) containing the cycles (1 2 3) and (3,5).
(a) List all the elements in H.
(b) How many left cosets of H in G are there?
Let H denote the smallest subgroup of S(5) containing the cycles (1 2 3) and (3,5)....
11-II Let H denote the smallest subgroup of S(5) containing the cycles (1 2 3) and (3,5 (a) List all the elements in H. b) How many left cosets of H in G are there?
(*) Let G be a group. Let G, G denote the smallest subgroup of G containing S = {xyr-ly-1: 2, YEG}. (The subgroup (G,G] is called the commutator subgroup of G.) (a) Show that u-zyr-?-?u= (u-cu)(u-yu)(u--xu)-1(u-yu)-1 for all 2, 4, U E G. Deduce that (G,G| 4G. (b) Show that the quotient group G/[G,G] is abelian. (c) If N 4G and G/N is abelian, show that (G,G] C N. (In other words, G/(G,G) is the largest abelian quotient of G.)...
Answer Question 5 . Name: 1. Prove that if N is a subgroup of index 2 in a group G, then N is normal in G 2. Let N < SI consists of all those permutations ơ such that o(4)-4. Is N nonnal in sa? 3. Let G be a finite group and H a subgroup of G of order . If H is the only subgroup of G of order n, then is normal in G 4. Let G...
Q. 5 5. Let H G be a subgroup and suppose that H,g2H.....gH are the distinet left cosets of H in G. Prove that gH - Hg for all g e G if and only if g.H Hg,, for all 2 sisr
(5) Let G be a group, and let H be a subgroup of G. Define a relation ~ on G as follows: X~ · y if x-ly E H. Prove that this is an equivalence relation, and that the equivalence classes of the relation are the left cosets of H.
3. a. Let H be a subgroup of a commutative group G. If every element h ∈ H is a square in H (i.e., h = k 2 for some k ∈ H), and every element of G/H is a square in G/H, then every element of G is a square in G. b. Let G be a group and H a subgroup with [G : H] = 2. If g ∈ G has odd order (i.e., ord(g) is odd),...
abstract algebra show your work 3. Let H be a subgroup of G with |G|/\H = 2. Prove that H is normal in G. Hint: Let G. If Hthen explain why xH is the set of all elements in G not in H. Is the same true for H.C? Remark: The above problem shows that A, is a normal subgroup of the symmetric group S, since S/A, 1 = 2. It also shows that the subgroup Rot of all rotations...
Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of H. (b) Prove or disprove that G/H isomorphic to G. G =C H={+1, i)
1. Let G - Z. Let H - {0,3,5,9) be a subgroup of (you do not need to prove this is a subgroup of G). Prove that G/l is a valid quotient group. Explain what the elements of G/H are and what the group operation is. 2. Let G be a group and H a normal subgroup in G. I E H for all IEG, then prove that G/H is abelian
2. problem 3. Let H be a normal subgroup of a group G and let K be any subgroup of G. Prove that the subset HK of G defined by is a subgroup of G Let G S, H ), (12) (34), (13) (24), (1 4) (23)J, and K ((13)). We know that H is a normal subgroup of S, so HK is a subgroup of S4 by Problem 2. (a) Calculate HK (b) To which familiar group is HK...