Determine the number of subgroups of (Z360,+). Also determine the number of automorphisms of (Z360,+).
Adv.
Prove that the homomorphic image of a cyclic (abelian) group is cyclic (abelian). Give an example to show that if the homomorphic image of a group is cyclic (abelian), the same may not hold for the group itself.
In (Z24,+) find all m∈Z24 such that ⟨21⟩=⟨m⟩.
Show that (Q+,.), the multiplicative group of positive rational numbers, is not a cyclic group.
Let f:G→G′ be a homomorphism of groups. Show that if H′⊴ , then f^{-1}(H^\prime ) \unlhd G and \ker f \subseteq f^{-1}(H^\prime). Further show that if f is surjective and H \unlhd G , then f(H) \unlhd G^\prime .
Show by an example that we can find groups G,H,K such that K \unlhd G and H \unlhd K, but H is not a normal subgroup of G.
Give an example of a group G and a subgroup H of G such that for some a \in G , aHa^{-1} \subsetneq H.
If a cyclic subgroup T of a group G is normal in G, then show that every subgroup of T is normal in G.
Show that if every cyclic subgroup of a group G is normal in G, then every subgroup of G is normal.
Suppose that H is the only subgroup of order \mid H \mid in a finite group G. Prove that H char G, and hence H \unlhd G .
Let H \le G \ and \ N(H) = \{g\in G \mid gHg^{-1}=H \}. Prove that
(i) N(H) \le G, (ii) H \unlhd N(H), (iii) If K \le G such that H \unlhd K, then K \subseteq N(H),
(iv) H \unlhd G iff N(H)=G.
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