# What finite groups have unique cyclic subgroups? [duplicate]

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Let $G$ be a finite group of order $n$. If $G$ is cyclic, we then know that are subgroups are cyclic are are unique. If $G=\langle x\rangle$, and $d|n$, then $\langle x^d\rangle$ is the unique cyclic subgroup of order $\frac{n}{d}$.

However, suppose we know that the group $G$ has a unique subgroup of order $d$ for any $d$ such that $d|n$. What else can we say about $G$? Does it have to be cyclic? Can it be factorizable over subgroup $H$ and $K$?

Any help is greatly appreciated!

## marked as duplicate by Derek Holt finite-groups StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 12 '17 at 16:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• See the posted solutions to this question – lulu May 12 '17 at 16:18
• Thanks. I don't know why I couldn't find that question earlier. – Scotty Vol May 12 '17 at 16:22
• This question could be a duplicate target. It is a duplicate of quite a few other questions already. – Jyrki Lahtonen May 12 '17 at 16:22

## 1 Answer

This condition implies that the Sylow $p$-subgroup of $p$ is unique and cyclic. This implies that $G$ is the direct product of its Sylow subgroups, so is cyclic.

• That's what I was thinking. I was hoping that there would be something that I was missing, but I wasn't 100% positive. Thank you. – Scotty Vol May 12 '17 at 16:20