Let G be a finite group, and suppose that for any two subgroups H and K of G either H⊆K or K⊆H. Prove that G is a cyclic group of prime power order.
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Show that the direct product of two infinite cyclic groups in not cyclic.
Determine the number of subgroups of (Z360,+). Also determine the number of automorphisms of (Z360,+).
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.
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