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The book I'm reading presents a proof for the following proposition. (Note: Sylows theorem has not been presented yet.)

Let $G$ be a group of order $2p$, where $p$ is a prime greater than $2$. Then $G$ is isomorphic to $Z_{2p}$ or $D_p$.

They first show that if there's an element of order $2p$ then it generates a cyclic group, which can be shown to be isomorphic to $Z_p$. If no such element exists, they show there can be elements of orders only $p$ and $2$. Letting $|a|=p$ and $|b|=2$, they show $ab=ba^{-1}$. They then make the following claim:

Since the multiplication table for all noncyclic groups of order $2p$ is uniquely determined by the relation $ab=ba^{-1}$, all ­noncyclic groups of order $2p$ must be isomorphic to each other.

How do I see that's true?

yoshi
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  • I'm unclear on the question. Is it "why are all groups with the same generators and relations isomorphic"? – ziggurism Nov 20 '17 at 00:21

1 Answers1

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Notice that $a$ and $b$ will generate the group, and assume you are given a relation $ab=ba^{-1}$. The elements of the group will be $\lbrace a^{k}, k=0,1,2 \cdots p-1 \rbrace$ and $\lbrace a^{q}, q=0,1,\cdots p-1\rbrace$.(It is important that you see this!). Now the relation $ab=ba^{-1}$ determines the multiplication, for example $ba^{q} ba^{k}=b(a^{q}b)a^{k}=b^{2}a^{-q}a^{k}=a^{k-q}$, $ba^{k} a^{q}=ba^{k+q}$, $a^{q}ba^{k}=b a^{k-q}$, $a^{q}a^{k}=a^{k+r}$

mich95
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  • what is the difference between your definitions of $a^k$ and $a^q$? Are you just saying that any number less than $p$ also generates $$? – yoshi Nov 20 '17 at 00:54
  • I'm unclear how it "generates the group". I only seeing that $ab$ will generate a group of size $2p$. – yoshi Nov 20 '17 at 01:00
  • I think I see: so, $ab$ is a group of size $2p$ with this relation. You computed all the types of multiplications that occured. Further this relation uniquely determines noncyclic groups of order $2p$. So this multiplication table is the only multiplication table for noncyclic groups of order $2p$. We can show $D_p$ satisfies this table too. Then we are done – yoshi Nov 20 '17 at 01:07