Question

Parts (A), (B) And (C) Are Unrelated. Let G Be A Group.

(A) Z(G) Is A Normal Subgroup Of G.
(B) Inn(G) Is A Normal Subgroup Of Aut(G)
(C)If H is the only subgroup of G of order |H|, then H is a normal subgroup of G. Hint. Something about Inn(G) might be useful.

Answer 1

In summary, (A) Z(G) is always a normal subgroup of G. (B) Inn(G) is also a normal subgroup of Aut(G). (C) The statement that H is a normal subgroup of G if H is the only subgroup of G of order |H| is false in general.

Step-by-step explanation:

Let's address each part of the question separately:

(A) Z(G) refers to the center of a group G, which is the set of elements that commute with every other element in G. It is proven that Z(G) is always a normal subgroup of G.

(B) Inn(G) is the set of inner automorphisms of G, which are automorphisms of the form x -> gxg^-1, for some fixed element g in G. Inn(G) is also a normal subgroup of Aut(G).

(C) If H is the only subgroup of G of order |H|, then H is a normal subgroup of G. This statement is false in general. Counterexamples can be found in groups that do not have a unique subgroup of a particular order, such as the symmetric group S_n for n >= 3.