Group Theory : A property related to the intersection of all subgroups

Question

I was solving a problem from the book by Herstein (problem 2.5.2) and came across the following solution (the question is given there itself): http://www.artofproblemsolving.com/community/q1h1083399p4773230

Can someone give me an example of such a situation? [Edit] In other words a group with the property that the intersection of its non-trivial subgroups is non-trivial as well. [/Edit]

Answer 1

An infinite group with this property is the following subgroup of the multiplicative group of complex numbers $$ G=\{z\in\Bbb{C}\mid z^{2^n}=1\ \text{for some natural number $n$}\}. $$ If $z\in G$ is $\neq1$, then $-1$ will also be a power of $z$. This is because if $k$ is the smallest non-negative integer with the property $z^{2^k}=1$ then we have $k>0$ (otherwise $z=1$). Consequently $W:=z^{2^{k-1}}$ satisfies both $w^2=1$ and $w\neq1$, so $w=-1$.

Therefore $-1$ belongs to the intersection of all non-trivial subgroups of $G$. Clearly $G$ is an infinite group, so this example works.

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We can use any prime number $p$ in place of $2$ above. In the more general case all the roots of unity of order $p$ will be contained in the intersection of all the non-trivial subgroups.

Answer 2

Sure, consider the cyclic group of order $4$. It has precisely $2$ non-trivial subgroups, the entire group, and the unique subgroup of order $2$. The intersection of these groups is the subgroup of order $2$.

You can generalize this to any cyclic group of order $p^n$, for $p$ a prime.