Give an example of a group G and a subgroup H of G such that for some a∈G,aHa−1⊊H.
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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.
Show that if N is a normal subgroup of finite index in a group G, and H is a subgroup of finite order in G such that [G:N] and ∣H∣ are relatively prime, then H⊆N.
Show that if N is a normal subgroup of finite order in a group G, and H is a subgroup of finite index in G such that [G:H] and ∣N∣ are relatively prime, then N⊆H.
Show that if G is a group with centre Z(G), and if G/Z(G) is cyclic, then G is abelian. Hence show that there does not exist any group G such that ∣G/Z(G)∣=p, where p is a prime number.
Prove that Q/Z is isomorphic to the multiplicative group of root of unity in C×.
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