Let $a$ and $b$ be elements of a group. If $|a|=10$ and $|b|=21$, show that $\langle a\rangle \cap \langle b\rangle = \{e\}$.
Let $a$ and $b$ belong to a group. If $|a|=24$ and $|b|=10$, what are the possibilities for $\left|\langle a\rangle \cap\langle b\rangle\right|$?
Show that the group of positive rational numbers under multiplication is not cyclic?
Prove that any group of order $4$ is abelian.
Prove that any group of order $5$ is cyclic.
Prove that any abelian group of order $6$ is cyclic.
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Q1) intersection ={e} Q2) |intersection| ? – tae ju Apr 10 '15 at 12:31
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For future question please learn how to format your question using $\LaTeX$. Also we'd like to hear your thoughts about the problems and what it is specificly that you're asking. A list like this of homework problems with no thoughts of your own is likely to be closed! – Christoph Apr 10 '15 at 12:49
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Some considerations:
For 1. and 2. use Lagrange's theorem and the fact that for subgroups $H$, $K$ of a group $G$, the intersection $H\cap K$ is a subgroup of both $H$ and $K$.
For 3. consider any cyclic subgroup $\langle q\rangle\subseteq\mathbb Q^+$ and construct an element in $\mathbb Q^+\setminus\langle q\rangle$.
For 4. consider the cases of $G$ being cyclic or not. If not, what are the orders of the group elements? (Lagrange's theorem again)
For 5. find the possible orders of the group elements.
For 6. find the possible group element orders again, now consider different possible cases and try to construct an element of order $6$.
Christoph
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