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Thread: Prove that the order of Z(G) is either <e> or G

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    Prove that the order of Z(G) is either <e> or G

    Let $\displaystyle G $ be a group of order $\displaystyle pq $, with $\displaystyle p $ and $\displaystyle q $ (not necessarily distinct) primes. Prove that the center $\displaystyle Z(G) = <e> $ or $\displaystyle Z(G) = G $.

    I started off by doing a proof by contradiction.

    Suppose that $\displaystyle Z(G) \neq <e> $, $\displaystyle Z(G) \neq G $

    Then $\displaystyle |Z(G)| $ divides either $\displaystyle p $ or $\displaystyle q $, which implies that $\displaystyle |Z(G)| = p $ or $\displaystyle |Z(G)| = q $

    We know that $\displaystyle Z(G) $ is a normal subgroup.

    Consider the quotient group $\displaystyle G/Z(G) $.

    Using Larange's Theorem, we know that $\displaystyle |G|/|Z(G)| $ (*).

    I broke it into two cases: 1: $\displaystyle |Z(G)| = p $, 2: $\displaystyle |Z(G)| = q $.

    If under case 1, (*) evaluates to $\displaystyle p $. Otherwise, in case 2, (*) evaluates to $\displaystyle q $.

    This is as far as I've gotten and I'm pretty stumped as how else to follow up after.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by crushingyen View Post
    Let $\displaystyle G $ be a group of order $\displaystyle pq $, with $\displaystyle p $ and $\displaystyle q $ (not necessarily distinct) primes. Prove that the center $\displaystyle Z(G) = <e> $ or $\displaystyle Z(G) = G $.

    I started off by doing a proof by contradiction.

    Suppose that $\displaystyle Z(G) \neq <e> $, $\displaystyle Z(G) \neq G $

    Then $\displaystyle |Z(G)| $ divides either $\displaystyle p $ or $\displaystyle q $, which implies that $\displaystyle |Z(G)| = p $ or $\displaystyle |Z(G)| = q $

    We know that $\displaystyle Z(G) $ is a normal subgroup.

    Consider the quotient group $\displaystyle G/Z(G) $.

    Using Larange's Theorem, we know that $\displaystyle |G|/|Z(G)| $ (*).

    I broke it into two cases: 1: $\displaystyle |Z(G)| = p $, 2: $\displaystyle |Z(G)| = q $.

    If under case 1, (*) evaluates to $\displaystyle p $. Otherwise, in case 2, (*) evaluates to $\displaystyle q $.

    This is as far as I've gotten and I'm pretty stumped as how else to follow up after.
    Lemma: If $\displaystyle G/\mathcal{Z}(G)$ is cyclic then, we have that $\displaystyle G$ is abelian.

    Proof: Let $\displaystyle x,y\in G$, then clearly $\displaystyle x\mathcal{Z}(G),y\mathcal{Z}(G)\in G/\mathcal{Z}(G)$. But, since $\displaystyle G/\mathcal{Z}(G)$ is cyclic we have that

    $\displaystyle x\mathcal{Z}(G)=g^m\mathcal{Z}(G)$ and $\displaystyle y\mathcal{Z}(G)=g^n\mathcal{Z}(G)$ for some $\displaystyle g\in G$. Thus, we have that $\displaystyle x=g^m\text{ } z_1,y=g^n\text{ }z_2$ for some $\displaystyle z_1,z_2\in\mathcal{Z}(G)$ (this is because cosets form a partition). And so

    $\displaystyle xy=g^m\text{ }z_1\text{ }g^n\text{ }z_2=z_2 \text{ }g^{m+n}\text{ }z_1=z_2\text{ }g^{n+m}\text{ }z_1=z_2\text{ }g^n\text{ }g^m\text{ }z_1$$\displaystyle =g^n\text{ }z_2\text{ }g^m\text{ }z_1=yx$ (since $\displaystyle z_1,z_2\in\mathcal{Z}(G)$). The conclusion follows. $\displaystyle \blacksquare$


    Now since $\displaystyle \mathcal{Z}(G)\lhd G$ we have, by Lagrange's theorem that $\displaystyle \left|\mathcal{Z}\left(G\right)\right|\mid\left|G\ right|=pq\implies \left|\mathcal{Z}\left(G\right)\right|=1,p,q,pq$. If it's the first or the last we're done. So, assume that it could be either $\displaystyle p$ or $\displaystyle q$. Then

    $\displaystyle G/\mathcal{Z}(G)=p,q$ and since all prime ordered groups are cyclic it follows from the lemma that $\displaystyle G$ is abelian and so $\displaystyle \mathcal{Z}(G)=G$ which contradicts the assumption that $\displaystyle \left|\mathcal{Z}(G)\right|=p,q$.

    The conclusion follows.
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