Question 45
We say that a group $(G, \cdot)$ has the property $(P)$ if, for any automorphism $f$ of $G$, there exist two automorphisms $g$ and $h$ of $G$ such that $f(x) = g(x) \cdot h(x)$, for any $x \in G$. Prove that:
a. Any group with the property (P) is commutative.
b. Any finite group of odd order has the property (P).
c. No finite group of order $4n + 2$, $n \in \mathbb{N}$, has the property (P).
a. Any group with the property (P) is commutative.
b. Any finite group of odd order has the property (P).
c. No finite group of order $4n + 2$, $n \in \mathbb{N}$, has the property (P).