Let G sub1 and G sub 2 be groups, with subgroups H sub 1 and H sub 2, respectivetly. Show that {(x sub 1, x sub 2) | H sub 1 is an element of H sub 1, x sub 2 is an element of H sub 2} is a subgroup of the direct product G sub 1 x G sub 2.

(1) Prove that if |G|=1365, then G is not simple.
(2) Assume that G is a nonabelian group of order 15. Prove that Z(G)=1. Use the fact that the group generated by "g" is less than or equal to C_G(g) for all "g" in G to show that there is at most one possible class equation for G.

Give an example of groups H_i, K_j such that H_1xH_2 is isomorphic to K_1xK_2 and no H_i is isomorphic to any K_j.
Let G be the additive group Q of rational numbers. Show that G is not the internal direct product of any two of its proper subgroups.
If G is the internal direct product of subgroups G_1 and G_2 show that G/G_

(1) Let G be cyclic of order pn, p prime. Let H,K < G. Show that either H C K or K C H.
(2) Let G 6 ≠ {e} such that it has no proper subgroups. Then G must be cyclic of prime order.
(3) If G is a group with order pq where p > q are primes and q does not divide p − 1, then G must be cyclic.
Please see the attached fi

A) Prove that if H is nontrivial normal subgroup of the solvable group G then there is a nontrivial subgroup A of H with A normal subgroup of G and A abelian.
b)Prove that if there exists a chain of subgroups G1<=G2<=.....<=G such that
G=union(from i=1 to infinity)of Gi and each Gi is simple, then G is simple
Part a of thi

Problem 1.
Let a,b be elements of a group G
Show a) the conjugate of the product of a and b is the product of the conjugate of a and the conjugate of b
b) show that the conjugate of a^-1 is the inverse of the conjugate of a
c)let N=(S) for some subset S of G. Prove that the N is a normal subgroup of G if
gSg^-1<=N for

Modern Algebra
Group Theory (CX)
Sylow's Theorem
Find all 3-Sylow subgroups of or, Sylow 3-subgroupsand 2-Sylow subgroup or, Sylow 2-subgroups of the symmetric group of degree 4, S4.

Thisquestion is concerned with subgroups ofthe group S5 of permutations on the set {1,2,3,4,5} , a group with 120 elements.
(a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6, and give examples of each of these.
Explain why this group does not have cyclic subgroups of any other order.
(8 marks
(b) By co