Let $G$ be a finite abelian group of order $n$ . Then choose the correct statement.

a) If d divides n, then there exist a subgroup of $G$ of order $d$

b) If d divides n, then there exist an element of $G$ of order $d$

c) If every proper subgroup of $G$ is cyclic then $G$ is cyclic.

d) If $H$ is a subgroup of $G$, then there exist a subgroup $N$ such that $\frac{G}{N} \cong H$.

for a) I know that the converse of the Lagrange's Theorem for abelian group is true.

for b) and d) Take $G = \mathbb Z_2 \times \mathbb Z_2\times \mathbb Z_2 \times \mathbb Z_2 $ has no element of order 4.

for c) take $G= \mathbb Z_2 \times Z_2$ , every subgroup of $G$ is cyclic but $G$ is not cyclic .

I would be thankful if someone checks my solution.

  • $\begingroup$ For (d) I have some doubts. $\endgroup$ – egreg Jun 20 '16 at 10:04
  • $\begingroup$ $G= \mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$, Is $G$ has a subgroup of order 4 $\endgroup$ – user120386 Jun 20 '16 at 10:14
  • $\begingroup$ How does your example help you for d)? $\endgroup$ – almagest Jun 20 '16 at 10:15
  • $\begingroup$ @almagest : Since $G$ has a subgroup of order 2 which i have taken as counter example, if $G$ has no subgroup of order 4., we get a ccounter examlple for d), I am not sure about d), i am trying to find a counnter example. $\endgroup$ – user120386 Jun 20 '16 at 10:23
  • $\begingroup$ @ egreg : what will you say about d) $\endgroup$ – user120386 Jun 20 '16 at 10:43

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