Quotient of direct sum of abelian groups

Question

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!

This question has been asked at least $10$ times on math.SE. — Martin Brandenburg, Feb 09 '16 at 07:49
They are equivalent, because $B \cong B'$ implies $A' \oplus B' \cong A' \oplus B$. Your question is not about equality. — Martin Brandenburg, Feb 09 '16 at 07:51
@helen: For a good link follow the Cancellation Theorem and on. — Mikasa, Feb 09 '16 at 07:53
@MartinBrandenburg omg, of course they were equivalent..Thank you!! — Sasha Mayer, Feb 09 '16 at 09:16

score 2 · Accepted Answer · answered Feb 09 '16 at 07:47

2

Let $A$ be the one-element group, and $A'=\mathbb{Z}_2$ and $B$ the direct sum of countably many copies of $\mathbb{Z}_2$.

answered Feb 09 '16 at 07:47

André Nicolas

1

A' can be arbitrary non-zero. – Martin Brandenburg Feb 09 '16 at 07:50
@André Nicolas: but is there simple reasonable explanation in case of finite groups? – Sasha Mayer Feb 09 '16 at 09:12
1

@helen: For finite (or finitely generated) Abelian groups, the Fundamental Theorem gives us cancellation. For finite groups in general, and more, there is the Krull-Schmidt Theorem, quite a bit harder. – André Nicolas Feb 09 '16 at 14:50

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