Prove that there is no non-trivial homomorphism from
(i) (Q,+) to (Z,+)
(ii) (Q,+) to (Zm,+)
(iii) (Q,+)to (Q∗,×), where Q∗=Q−{0}.
Adv.
Prove that there is no surjective homomorphism from
(i) Z16×Z2 to Z4×Z4 (ii) Z4×Z4 to Z8 .
If ϕ is a homomorphism from Z30 onto a group of order 5, determine ker ϕ .
Let G be an abelian group and n a positive integer. Define f:G→G by f(x)=xn . Show that f is an endomorphism of G. Further if G is finite such that n and |G| are coprime, show that f is an automorphism of G .
If a group G contains an element a with exactly two conjugates, then show that G has a normal subgroup N≠{e} .
Let H be a proper subgroup of a finite group G . Show that G contains at least one element that is not in any conjugate of H .
Let G be a non-abelian group of order 21. Prove that Z(G)={e} . Hence show that there is at most one possible class equation for G.
Let G be an abelian group and H,K two finite subgroups of G of orders m and n, respectively. Prove that G has a subgroup of order LCM(m,n) .
(a) Show that the center of a group G is a characteristic subgroup of G.
(b) Prove that characteristic subgroups are normal. Give an example of a normal subgroup which is not characteristic.
(c) Show that if H is the only subgroup of order n in G, or if H is the only subgroup of index k in G, then H is characteristic in G.
Prove that there are no simple groups of orders 42, 56, 96, 108, 200, 1986.
Let G be a group of order 1575. Prove that if H is a normal subgroup of order 9 in G, then H≤Z(G).
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