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Let $G$ be an abelian group and $H$ subgroup of $G$. Suppose that:

(i) $H$ has a complement in $G$.

(ii) $K$ is a subgroup of $G$ and K is isomorphic to $H$

Is there a complement of $K$ in $G$? If yes, what is the relation of complements of $H$ and complements of $K$?

Thanks in advance.

Luyen Le
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1 Answers1

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Not true. Let $G$ be the group $\mathbb Z/4 \mathbb Z \oplus \mathbb Z/2\mathbb Z$. Let $H$ be the second factor in the direct sum, which is isomorphic to $\mathbb Z/2 \mathbb Z$. Let $K$ be the order $2$-subgroup of $\mathbb Z/4 \mathbb Z$, considered as a subgroup of the first factor of the aforementioned direct sum.

  • You mean $H$, not $K$, right? – BIS HD Oct 29 '13 at 14:11
  • Indeed, corrected. Thanks. –  Oct 29 '13 at 14:11
  • Thanks @Doldrums In case: $G$ is a finite direct sum $\mathbb{Z}/p\mathbb{Z}$. It means $G:=\mathbb{Z}/ p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}\oplus\cdots\oplus \mathbb{Z}/p\mathbb{Z}$. Is the question true? – Luyen Le Oct 29 '13 at 14:24
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    @user57179 In that case, $G$ is a $\mathbb{Z}/p\mathbb{Z}$ vector space, and all subgroups of $G$ are too. Then all subgroups are complemented. – Daniel Fischer Oct 29 '13 at 14:29