Defn: Let  H be a subgroup of G.  NG(H)={g∈ G | gHg-1=H } is called the normalized of H in G. a) let H be a subgroup of G, Prove that  H  is a normal subgroup of  NG(H).

If H is a subgroup of G, then H is a normal subgroup in G if and only if H × A2021 is a normal subgroup in G × S2021. Please explain the steps in detail, thank you in advance.

Let H be a subgroup of G, and define its normalizer as N(H) := {g G: gHg¹ = H}. (i) Show that N(H) is a subgroup of G. (ii) Show that the subgroups of G that are conjugate to H are in one-to-one correspondence with the left cosets of N(H) in G.

Consider group G and its subgroup H (H ≤ G): [G : H] = 2. Prove why H is a normal subgroup of G.