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 .
Adv.
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.
If G is a finite group and N \unlhd G such that \mid N \mid and [G:N] are coprime, show that for any a \in G ,a ^{\mid N \mid } = e \Rightarrow a \in N.
If H \le G, such that x^2 \in H for all x \in G. Prove that H \unlhd 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 \mid H \mid are relatively prime, then H \subseteq 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 \mid N \mid are relatively prime, then N \subseteq H.
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