An isomorphism between Groups of order $p^2$

Question

Let $G$ a group of cardinality $p^2$ such that every element of $G$ is of order different than $p^2$.

Let $a,b\in G\setminus\{e\}$ such that $b\notin\langle a\rangle$. Of course the order of $a$ and $b$ is $p$.

We consider the mapping $\varphi:(Z/pZ)^2\to G, (\overline{k},\overline{\ell})\to a^kb^\ell$.

Of course we can show easily that this mapping is well-defined and homomorphism of groups.

For the surjectivity of $\varphi$ i have the following indication: Show that $Im(\varphi)$ contain at least $p+1$ element.

If that is true, we can conclude easily that $Im(\varphi)=G$. but the question here why $Im(\varphi)$ contain at least $p+1$ element. Thank you for your help in advance.

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A group of order $p^2$, which has no element of order $p^2$ is abelian, but not cyclic. So it must be isomorphic to $C_p\times C_p$. See for example this duplicate, or this one. — Dietrich Burde, Sep 15 '24 at 15:32
Use \langle and \rangle for delimiters, not < and >. — Arturo Magidin, Sep 15 '24 at 19:00
There are false statements in the question. Let $p=4$. Let $G$ be the group of symmetries of a regular octagon, a group of order $p^2$ with no element of order $p^2$. It has elements, other than the identity, which are not of order $p$. — Gerry Myerson, Sep 18 '24 at 00:16

score 5 · Accepted Answer · answered Sep 15 '24 at 11:29

5

Note that there is a problem with your mapping $\varphi$. It will be an homomorphism provided you know that $G$ is abelian, otherwise it might not be true.

This problem asides, to answer your question, the image of $\varphi$ contains $1,a,\ldots,a^{p-1}$ and $b$ by definition. Using your assumptions on $a$ and $b$, you might be able to show that these elements are parwise distinct.

answered Sep 15 '24 at 11:29

GreginGre

16,641

1

Yes of course, in this problem we proved that G is abelien by the class equation. – aymen orton Sep 15 '24 at 11:31
@aymenorton Then you should mention that in your question. – jjagmath Sep 15 '24 at 19:43

An isomorphism between Groups of order $p^2$

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