Only two groups of order 10.

Question

I have shown a group of order 10 has a cyclic subgroup of order 5. But now I am stuck. Any hints?

Then what can I do to get a better answer which I understand? — Lin, Feb 04 '16 at 22:20
I would comment on the answers there asking about the specifics that you don't understand. — Stella Biderman, Feb 05 '16 at 00:22

Stella Biderman · Answer 1 · 2016-02-04T22:16:31.130

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Observe that $D_{10}$ and $C_{10}$ are both groups of order $10$. You know that there is a subgroup of order $5$, so what does that tell you about the possible arrangements of the rest of the elements of the group? Consider assuming there is an element of order 10, and then try assuming there is no element of order 10. Each assumption should be enough to prove $G$ is one of the two groups above. You know there is a group of order 2 inside of $G$ as well, and $(2,5)=1$

edited Feb 04 '16 at 22:16

answered Feb 04 '16 at 22:09

Stella Biderman

31,155

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By $C_5$ is meant the dihedral group here? (If so I think it is usually denoted by a $D$ with subscript. – coffeemath Feb 04 '16 at 22:11
I mean the cyclic group? – Stella Biderman Feb 04 '16 at 22:14
I need to show C_10 and D_10 are the only 2 groups of order 10 – Lin Feb 04 '16 at 22:14
Yes. $D_10=Z/10Z$ – Stella Biderman Feb 04 '16 at 22:15
But $\mathbb{Z}/10\mathbb{Z}$ is already the cyclic group of order $10$. – Tobias Kildetoft Feb 04 '16 at 22:15
Wow yes I'm silly – Stella Biderman Feb 04 '16 at 22:16
What can I say about the group of order 2? – Lin Feb 04 '16 at 22:36

Only two groups of order 10.

1 Answers1