Checking if Two Groups of Order 100 are Isomorphic

Question

Is $D_5 \times \mathbb Z/10\mathbb Z$ isomorphic to $D_{10} \times \mathbb Z/5\mathbb Z$?

I am calculating that each of these order 100 groups has the same maximal element order of 10, and the same number of elements of orders 1 (1 element), 2 (11 elements), and 5 (24 elements), hence 10. However, example 4 on page 158 of Gallian's Contemporary Abstract Algebra text seems to suggest that I must have done something wrong.

Apparently these groups are indeed isomorphic: see Corollary 1.3 of these notes by Keith Conrad. — Greg Martin, Jun 29 '24 at 09:21

Dietrich Burde · Accepted Answer · 2024-06-30T09:25:16.297

4

We have $D_{10}\cong D_5\times C_2$ and $C_{10}\cong C_5\times C_2$, because $\gcd(5,2)=1$. This yields $$D_5\times C_{10}\cong D_5\times C_2\times C_5\cong D_{10}\times C_5.$$

For the first isomorphism see this post, for example.

Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.

edited Jun 30 '24 at 09:25

answered Jun 29 '24 at 10:01

Dietrich Burde

140,055

This answer would be perfect if it explained or included a link explaining the first isomorphism. – Johnny Apple Jun 30 '24 at 02:17
I have included a link. Also, Greg gave a link in his comment. – Dietrich Burde Jun 30 '24 at 07:56

Checking if Two Groups of Order 100 are Isomorphic

1 Answers1