# Tagged Questions

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

47 views

47 views

111 views

### Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
35 views

### Let $F = VN$ that $V \cap N = 1$ . Let $L = N_{G}(V)$. $(\vert N \vert , \vert F/N \vert) =$?

Let $G$ be a soluble group and $A$ be a minimal normal subgroup of $G$,where $A$ is an elementary abelian group of prime power order. Let each chief factor of $G/A$ has order $4$ or a ...
23 views

### Find the intertwiner of two equivalent group representations

Let $G$ be a finite group. Let $$\{A(g)\mid g \in G\} \text{ and }\ \ \ \{B(g) \mid g \in G\}$$ be two equivalent unitary matrix representations of $G$. How do I find a unitary intertwining matrix $S$...
53 views

### Let $Q$ be a $p$ -group and let $N$ be a nontrivial, elementary abelian normal subgroup of $Q$

Let $Q$ be a $p$ -group and let $N$ be a nontrivial, elementary abelian normal subgroup of $Q$ which has a complement $X$ in $Q$. If $Q = \langle y \rangle X$ for some element $y$, then ...
60 views

### Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
67 views

### Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
60 views

### Short exact sequence of groups

Let $1\to N\xrightarrow{\text{i}} G\to H\to 1$ be a short exact sequence of groups. Let $f:G\to N$ be a group homomorphism such that $fi: N\to N$ is an isomorphism. What are some "good" things we can ...
32 views

### Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
69 views

I have this question in my home assignment. I contains two parts and I don't quite understand what is the difference between them.The question is: Let $n > 2$ be an integer such that $(\mathbb{Z}/n\... 0answers 59 views ### Showing the existence of$O^{\pi}(G)$Assume G is a finite group. I am trying to show the existence of$O^{\pi}(G)$, the unique normal subgroup of G minimal such that$G/O^{\pi}(G)$is a$\pi$-group, i.e. a group whose order is divisible ... 0answers 32 views ### How to find the power of generator defined over finite field ,$\mathbb F_{2 ^m}$? List item Actually ,I am trying to execute the algorithm to find the power of generator of field(group) as shown in table of attached file but when k=7 and onward ,I could not understand what is ... 0answers 34 views ### If$C_A(F(A)) \le F(A)$and$C_B(F(B)) \le F(B)$, then this also holds for$AB$if$A,B \unlhd G$. Let$A, B \unlhd G$be normal subgroups of a finite group$G$such that $$C_A(F(A)) \le F(A) ~\mbox{ and }~ C_B(F(B)) \le F(B).$$ where$F(G)$denotes the Fitting Subgroup of$G$. I want to show ... 0answers 49 views ### does minimality condition imply normal p-sylow subgroup > Assume that$G$is a finite group, and$p$is a prime number dividing the order of$G$. Let$\cal C=\cal C(H)$be the following condition : "$H$is a normal subgroup of$G$and$|G/H|$is coprime to$...
If $X,Y$ are two subsets of some group $G$, then $$[X,Y] := \langle [x,y] : x \in X, y \in Y \rangle$$ is the commutator subgroup generated by $X$ and $Y$ (where $[x,y] := x^{-1}y^{-1}xy)$. Are ...
Suppose that $G$ is a finite group and we have a unitary irreducible representation $\rho:G\rightarrow \hom V$. Suppose we fix a basis $\{e_i\}_{i\geq 1}$ of $V$ and with respect to this basis we have ...