Question 16
A ring $(A,+, \cdot)$ has property (P) if $A$ is finite and the multiplicative group of its invertible elements is isomorphic to a subgroup of the additive group $(A,+)$ different from $\{0\}$. Show that:
a. If a ring has property $(\mathrm{P})$, then its number of elements is even.
b. For infinitely many natural numbers $n$, there exist rings with exactly $n$ elements which have property $(\mathrm{P})$.
a. If a ring has property $(\mathrm{P})$, then its number of elements is even.
b. For infinitely many natural numbers $n$, there exist rings with exactly $n$ elements which have property $(\mathrm{P})$.