# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

28 views

### Cyclic permutation group example ($n>1$)

I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.
79 views

### A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic

Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e)$. Prove that $G$ must be cyclic. Also what can be said about $o(G)$ ? I'm ...
29 views

### Representation of $A_5$

Can someone give me a proper reference (a book probably)for how a 3 dimensional representation of the Alternating group $A_5$ is related to the reflection group $H_3$ or the Icosahedral group ? Thanks
57 views

30 views

### Showing order of $Z (G) =1$ or $pq$

Question : If $|G|=pq$ where $p$ and $q$ are primes that are not necessarily distinct. Prove that the order of $Z (G) =1$ or $pq$. Showing the order is $pq$ is trivial. I unsure how to start ...
55 views

### $|G| = ?$ if its subgroups are $\{e\}$, $G$ itself, and a subgroup of order $7$?

Suppose that a cyclic group $G$ has exactly three subgroups: $G$ itself, $\{ e \}$, and a subgroup of order $7$. What is $|G|$? What can you say if $7$ is replaced with $p$ where $p$ is prime? I ...
49 views

### Order of a center of a group is prime order

Question : Suppose that $G$ is a non-abelian group of order $p^{3}$ where $p$ is prime and $Z (G) \neq \{e\}$. Prove that $|Z (G)| =p$. Any useful hint to this question is appreciated. ...
Let $G$ a group and $\chi$ a linear character, p$\ne$2 a prime number with $|G|$=$p^a$$m$ and $(p,m)=1$. Show that $\chi$ is p-rational if and only if p doesn't divide the order of $\chi$ (as element ...