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Question:
Give an example of a group $G$ such that $|G| = p^k$ for some prime $p$ and some integer $k > 1$, where $|g| = p$ for every $g \in G \setminus {\epsilon}$.
My Reasoning :
I wasn't too sure how to approach this question. Would the group $\mathbb{Z}_2 \times \mathbb{Z}_2$ be such a group? I wasn't too sure because the operation of this group is addition and not multiplication so I can't take powers of elements, which is what this question asks. for example : $(1,0)^2 = (1,0)$ $\neq$ $(0,0)$. I also thought that I can just take the general Klein group where: $V = \{1, a,b,c\mid a^2=b^2=c^2\}$. Are these two groups good examples? if not, what other group can I pick?

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    $\begingroup$ If you use multiplicative notation, $a^n$ is a shorthand for an n-fold product $a \cdots a$. If you use additive notation, you write $n\cdot a$, which then means $a + \cdots + a$ ($n$ times). If the problem statement uses multiplicative notation, it does not restrict you to it - use can use additive notation too, just translate carefully products to sums and powers to products. $\endgroup$
    – Al.G.
    Mar 25 at 19:00
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    $\begingroup$ If you follow ALG's remark, both your examples are valid for your problem. [Note also that the two examples you give are isomorphic groups, just keep in mind the operations are multiplication in one of them and addition in the other. $\endgroup$
    – coffeemath
    Mar 25 at 19:16
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    $\begingroup$ $g^n$ just means "the group operation being applied to $g$ $n$ times"; or, more visually, $\underbrace{g\cdot g\cdot ...\cdot g}_{n\text{ times}}$. It doesn't matter if the group operation is addition or multiplication or something else entirely. $\endgroup$
    – IAAW
    Mar 25 at 20:31
  • $\begingroup$ Those two groups are actually... one, i.e. they are isomorphic. More generally, $C_p\times C_p$ has that property, for every prime $p$. $\endgroup$
    – tronk
    Mar 25 at 21:50
  • $\begingroup$ To make this more interesting, try to find a non-abelian group with this property. $\endgroup$ Mar 26 at 7:53

1 Answer 1

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More generally: $$G=C_p\times C_p$$ where $C_p$ is the cyclic group of order $p$ (in multiplicative notation), has those properties: it has order $p^2$ and every nontrivial element has order $p$. Your example (one, actually, as $\Bbb Z_2\times \Bbb Z_2\cong V$) is just the special case for $p=2$. So, yes, it is a good example.

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