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A basis $B$ is a set of elements in $G$ such that every element in $G$ can be expressed as a multiplication of powers of elements in $B$, and if $a_1,..., a_n \in B$ and $a_1^{l_1}\cdots a_n^{l_n}=e$ then $a_1^{l_1}=\cdots =a_n^{l_n}=e$.
How can I prove that an abelian p group $G$ has a basis? I thought about proving that some quotient group of $G$ had a basis and then using a homomorphism to prove the same for $G$ but I could not prove it that way either. Any ideas?

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If your $p$ group is abelian, then this follows immediately from the Fundamental Theorem for Abelian Groups. If not, every $p$ group has a nontrivial center, so induction is your friend.

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  • $\begingroup$ It had been staring me right in the face this whole time! $\endgroup$
    – GuPe
    Mar 22, 2017 at 23:57

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