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How do we find the total no. Of subgroups of a group, in Zn,Sn,An,Dn? And there's this theorem that if 'a' belongs to Group G,then is a subgroup of G. Does that mean that we already hve available subgroups equal to the no. Of elements in the group.

Rajni
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  • "And there's this theorem that if 'a' belongs to Group G,then is a subgroup of G". I don't recall ever reading such nonsense. The conclusion you wish to draw from it is false, since $\Bbb Z/5\Bbb Z$ has just two subgroups. –  Jul 11 '16 at 06:22
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    You may be thinking of "If $a$ is an element of a group $G$, then the set generated by $a$ (labelled as $\langle a\rangle$ and defined as ${a^n~:~n\in\Bbb Z}={\dots,a^{-2},a^{-1},e,a^1,a^2,\dots}$ using multiplicative notation) is a subgroup of $G$" The problem with your conclusion is that not every choice of $a$ determines a unique subgroup. A subgroup could have multiple generators. – JMoravitz Jul 11 '16 at 06:29
  • I recall, perhaps incorrectly, that the problem of finding the number of subgroups of $S_n$ is very hard. For $\mathbb{Z}_n$, the number of subgroups is the number of divisors of $n$. – André Nicolas Jul 11 '16 at 06:39
  • Okay...if i were to find the no. Of subgroups of a certain order say 7 of a group of order 168. What is the procedure to be followed? – Rajni Jul 11 '16 at 08:42

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