In (Z24,+) find all m∈Z24 such that ⟨21⟩=⟨m⟩.
Adv.
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′⊴G′ , then f−1(H′)⊴G and kerf⊆f−1(H′). Further show that if f is surjective and H⊴G, then f(H)⊴G′ .
Show by an example that we can find groups G,H,K such that K⊴G and H⊴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∈G,aHa−1⊊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 ∣H∣ in a finite group G. Prove that H char G, and hence H⊴G .
Let H≤G and N(H)={g∈G∣gHg−1=H}. Prove that
(i) N(H)≤G, (ii) H⊴N(H), (iii) If K≤G such that H⊴K, then K⊆N(H),
(iv) H⊴G iff N(H)=G.
If G is a finite group and N⊴G such that ∣N∣ and [G:N] are coprime, show that for any a∈G,a∣N∣=e⇒a∈N.
If H≤G, such that x2∈H for all x∈G. Prove that H⊴G and G/H is abelian.
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