Hw6 Problem 6

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Problem 6

Prove or disprove: If every proper subgroup of a group $G$ is cyclic, then $G$ is also cyclic.

Solution

Consider the Klein 4 Group. The group itself is not cyclic, but every subgroup is cyclic.

Subgroups of $V$ include $\{e,a\}, \{e,b\}, \{e,c\}, \{e\}$ which are each generated by a single element in $V$.

$\langle a \rangle = \{e,a\}$
$\langle b \rangle = \{e,b\}$
$\langle c \rangle = \{e,c\}$
$\langle e \rangle = \{e\}$