(i) Let G be a p-primary abelian group, where p is prime. If...

(i) Let $G$ be a $p$-primary abelian group, where $p$ is prime. If $(m, p)=1$, prove that $x \mapsto m x$ is an automorphism of $G$. (ii) If $p$ is an odd prime and $G=\langle g\rangle$ is a cyclic group of order $p^{2}$, prove that $\varphi: x \mapsto 2 x$ is the unique automorphism with $\varphi(p g)=2 p g .$

(i) Let $G$ be a $p$-primary abelian group, where $p$ is prime. If $(m, p)=1$, prove that $x \mapsto m x$ is an automorphism of $G$.
(ii) If $p$ is an odd prime and $G=\langle g\rangle$ is a cyclic group of order $p^{2}$, prove that $\varphi: x \mapsto 2 x$ is the unique automorphism with $\varphi(p g)=2 p g .$

Key Concepts

p-primary Abelian Group

A p-primary abelian group is an abelian group in which every element has order a power of a fixed prime p. This concept is fundamental when analyzing group homomorphisms in contexts where the prime divisors of element orders control the group's structure.

Automorphism

An automorphism is a bijective homomorphism from a group to itself. It is an isomorphism that preserves the group operation, making the structure of the group invariant under the mapping.

Multiplication Map by an Integer

For abelian groups, a map defined by sending each element x to m·x is a group homomorphism. When the integer m is such that it is relatively prime to the prime that controls the orders of the elements (i.e., (m, p) = 1 in a p-primary group), this multiplication map becomes an automorphism because it is both injective and surjective.

Coprimality and Units

Coprimality in this context means that the integer m has no common prime factors with the prime p. This is equivalent to m being a unit modulo p, which ensures that multiplying by m is an invertible operation in the modular arithmetic framework underlying the structure of p-groups.

Cyclic Groups and Generator Actions

In cyclic groups, every element is expressed as a power (or multiple) of a single generator. Automorphisms of cyclic groups are entirely determined by their effect on a generator. When an automorphism is specified by its action on a generator (or a specific element like p·g in a cyclic group of order p²), exploring its uniqueness involves verifying that no other automorphism can produce the same effect on that generator.

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