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I have a question regarding group theory,

Let G be a finite group and a,b elements in G.

  1. if o(a) = 2, o(b) = 3 then o(ab) = 6.
  2. o(a) = o(b) = o(ab) = 2 then ab=ba

I am new to group theory and can't really point out how to approach this. Any tips will be gladly appreciated!.

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    Do you want to find examples of elements $a,b$ with this properties ?? – 2017-02-09

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The second one is a great classic here, $b a = b^{-1} a^{-1} = (a b)^{-1} = a b$, as $x^{2} = 1$ is equivalent to $x = x^{-1}$.

The first one is false in a general group (think of $S_{3}$) but true in an abelian group - is this what you meant?

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    Thanks! I havent seen the Sn group set. Could you try and find a different group set if exists from modulo,permutations and such. – 2017-02-09
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    @Tom.A, two questions first. In question 1, is $G$ assumed to be abelian? And then, have you seen permutations already? Because $S_{3}$ is just the group of permutations on the set $\{ 1, 2, 3 \}$. – 2017-02-09
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    Oh ok.. so its all clear concerning Sn. The group is not assumed to be abelian. Only finite. – 2017-02-09
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    Then just take any element $a$ of order $2$ and $b$ of order $3$ in $S_{3}$. There are of course no elements of order $6$ in $S_{3}$, as it is non-abelian and thus non-cyclic. – 2017-02-09