Abelian Groups

Classify all abelian groups of order 100.

Prove 1.7.

Let $G$ be an abelian group.

1. Prove that $G^{n}=\{x^{n}\mid x\in G\}$, where $n$ is a positive integer, is a subgroup of $G$.

2. Suppose also that $G$ is finite and $p$ is a prime that divides $|G|$. Using the map $\phi:G\to G^{p}$ where $\phi(x)=x^{p}$, prove that $G^{p}$ is a proper subgroup of $G$.(Hint: consider Cauchy’s theorem (1.31).)

Suppose $p_{1},p_{2},\dots,p_{n}$ are distinct primes. How many distinct abelian groups of order $p_{1}p_{2}\cdots p_{n}$ are there, up to isomorphism?

What is the smallest positive integer $n$ such that there exist two non-isomorphic abelian groups of order $n$? Write down the isomorphism class of these two groups.

Write out all isomorphism classes of an abelian group of order $p^{4}$, where $p$ is a prime.

Let $G$ be an abelian group of order 16. Suppose there exists $a,b\in G$ such that $|a|=|b|=4$ and $a^{2}\neq b^{2}$. Determine the isomorphism class of $G$.

How many abelian groups of order 16 have the property that $x^{4}=e$ for all $x$ in the group, up to isomorphism?