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I am reading the direct products and here is a theorem that I could't understand its serve well:

A group of order 30 is ismorphic to one of

$C_{30}, C_5\times D(3), C_3\times D(5)\qquad \qquad,$ or $\qquad D(15)$

Which type of groups are $C_{30}, C_5, C_3 $? And I also couldn't follow the reasoning in this example that the book gave after that theorem:

The group with presentation $$ \big <x,y : x^{15}=1=y^2, yxy^{-1}=x^4 \big > $$ is isomorphic to the direct product $C_3\times D(5)$

Improve this question
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  • $\begingroup$ $C_n$ denotes the cyclic group of order $n$. $\endgroup$ – Prasun Biswas Dec 21 '17 at 0:49
  • $\begingroup$ You should start from recalling that $D_n$ is not an abelian group if $n\geq 3$, then compare the multiplication tables of $C_3\times D_5$ and the group presented by such relations (this is a pretty inefficient approach, indeed, but it surely works). $\endgroup$ – Jack D'Aurizio Dec 21 '17 at 0:49
  • $\begingroup$ You can find more detail atlas of finite group:brauer.maths.qmul.ac.uk/Atlas $\endgroup$ – 1ENİGMA1 Dec 21 '17 at 6:29
  • $\begingroup$ Your first question answer is there:math.stackexchange.com/questions/569226/… $\endgroup$ – 1ENİGMA1 Dec 21 '17 at 6:36

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