2. Suppose an element a E G has order 2. Prove that (a) is a normal subgroup of G if and only if a is in the center of G.

Prove that every group of order 78 has a normal subgroup of order 39.

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.