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.

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132 views

Non-abelian groups of order $p^2q$

Let $G$ be a non-abelian group of order $p^2q$ and $p> q$. i) $p \equiv 1 \mod q$ or $p \equiv -1 \mod q$; ii) The number of elements of order $pq$ in group $G$ is $0$ or $p(p-1)(q-1)$.
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60 views

Assume simply connectivity without loss of generality

Let $X$ a connected Riemann surface and $G$ a finite group that acts faithfully and holomorphically on $X$. Further, let $x \in X$ a non-trivially stabilized point (we know these points are discrete), ...
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1answer
147 views

Subgroups of order 8 in the quasidihedral group of order 16

Why are there only $3$ subgroups of order $8$ in the quasidihedral group $QD_{16}$ of order $16$? (I am not interested in drawing the lattice of subgroups, but rather an argument convincing one that ...
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0answers
156 views

Class of $p$-supersolvable group is saturated formation.

I want to show that the class of $p$-supersolvable groups is a saturated formation. I only have the definition of $p$-supersolvable. What do I do? A group is $p$-supersolvable iff every chief factor ...
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1answer
282 views

Irreducible representations over $\Bbb R$

How to prove that all irreducible representations over $\mathbb{R}$ of finite abelian group have dimension 1 or 2?
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1answer
105 views

If the group $Q$ is $\langle x, y \mid x^{9}=y^{3}=1, x^{y}=x^{-1}\rangle$, what is the subgroup of $Q$ generated by its elements of order dividing 3?

Let $Q=\langle x, y \mid x^{9}=y^{3}=1, x^{y}=x^{-1}\rangle$ and suppose we define a subgroup $‎\Omega‎_1(Q)$ to be the subgroup of $Q$ generated by all elements in $Q$ of order dividing 3. Can one ...
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3answers
554 views

Classifying the groups of order $2013$ (up to isomorphism)

Let $G$ be a group such that $|G|=2013$, how would you classify, up to isomorphism, all groups $G$?
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2answers
47 views

Normal subgroups of $\langle(123),(456),(23)(56)\rangle$

Let $G$ be a subgroup of the symmetric group $S_6$ given by $G=\langle(123),(456),(23)(56)\rangle$. Show that $G$ has four normal subgroups of order 3. I may be missing something, but I can ...
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0answers
50 views

How to build $\operatorname{Hom}(\mathbb{Z}_p^*,\mathbb{Z}_{pq}^*)$ without solving DLP?

For given two distinct primes p and q, I want to construct all homomorphisms from the multiplicative group $\mathbb{Z}_p^*$ to the multiplicative group $\mathbb{Z}_{pq}^*$. Thanks to Jyrki ...
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0answers
116 views

trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...
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2answers
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Can the semidirect product of two groups be abelian group?

while I was working through the examples of semidirect products of Dummit and Foote, I thought that it's possible to show that any semdirect product of two groups can't be abelian if the this ...
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151 views

How useful are geometric aspects when studying finite groups?

My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of ...
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2answers
1k views

Order of products of elements in a finite Abelian group

We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$. How I approached this question was by saying let ...
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2answers
70 views

How to decide whether a p-subgroup of some sporadic groups is cyclic?

Suppose that H is a subgroup of some sporadic groups (say convey groups Co1, Co2, etc.) and $|H|=p^k$ for some prime p and integer k>1, how to determine whether H is cyclic?
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0answers
96 views

Supersolvable group and nilpotent maximal subgroup

$G$ is a supersolvable group, $|G|=pq^{b}$ ($p$ and $q$ are different prime numbers). $Q$ is a $q$-Sylow subgroup of $G$, $Q=\langle x\rangle$ and $\operatorname{Cor}_{G}(Q)=\langle x^{q}\rangle$. If ...
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2answers
115 views

Are homomorphisms into PGL related to the Schur multiplier?

I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...
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2answers
66 views

Application of Cauchy theorem to prove normality of a subgroup

Let $G$ be a group $o(G)=pq$, where $p,q$ are both distinct prime numbers. Let $H<G$ be a subgroup of $G$ and $o(H)=p$. I want to show that $H$ is normal in $G$. My argument goes as follows. First ...
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1answer
219 views

Generalizations of fitting subgroup

The Fitting subgroup of a group $G$ has two generalizations: the generalized Fitting subgroup $F^*(G)$ of Bender and $\tilde F(G)$ of Schmid. The latter is defined by $\tilde F(G)/\Phi(G) = ...
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2answers
265 views

if $G$ is a group , $H$ is a subgroup of $G$ and $g\in G$ , Is it possible that $gHg^{-1} \subset H$? [duplicate]

if $G$ is a group , $H$ is a subgroup of $G$ and $g\in G$ , Is it possible that $gHg^{-1} \subset H$ ? this means , $gHg^{-1}$ is Proper subgroup of $H$ , we know that , $H \cong gHg^{-1}$ , so if ...
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1answer
84 views

What is wrong with my thinking, simple groups order $168$

How many elements of order $7$ are there in a simple group of order $168$? I will work on this more but I have seen some solutions out there. My only question is regarding what is wrong what my ...
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2answers
142 views

About free group and kernel of homomorphism

I'm now reading textbook in group theory but couldn't understand its briefy explanation below "Let $G=<a,b\mid a^4=e,b^2=e,bab^{-1}=a^{-1}>, S=\{a,b\},F(S)$ be a free group and $N$ be the ...
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0answers
45 views

Maximal subgroups of $G=Z_{3}\ltimes Q_{8}$

Let $G=Z_{3}\ltimes Q_{8}$. How can find Maximal subgroups of $G$ ? $$Q_{8}$$ is Quaternion group of order of 8 and $$Z_{3}$$ is cyclic group of order 3
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1answer
89 views

Is there other homomorphisms from $\mathbb{Z}_q^*$ to $\mathbb{Z}_{pq}^*$?

For given two distinct primes p and q, is there other homomorphisms from the multiplicative group $\mathbb{Z}_q^*$ to the multiplicative group $\mathbb{Z}_{pq}^*$, except the following two maps: ...
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1answer
106 views

Is there non-trivial homomorphism from $\mathbb{Z}_q^*$ to $\mathbb{Z}_p^*$?

For two distinct primes $q$ and $p$, is there non-trivial homomorphism from $\mathbb{Z}_q^*$ to $\mathbb{Z}_p^*$? Here, $\mathbb{Z}_q^*$ and $\mathbb{Z}_p^*$ mean the multiplication groups with ...
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2answers
389 views

Presentation of a non-abelian group of order $pq$.

What is the presentation of the non-abelian group of order $pq$ where $p$ and $q$ are primes and $q\mid(p-1)$? Thanks in advance.
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1answer
420 views

Books to understand the construction of all groups of a specific order

The algorithms introduced by Besche–Eick (1999) were used to construct (or count) the groups of order up to 2000 in Besche–Eick–O'Brien (2002), yet I find the algorithms somewhat inaccessible. How ...
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1answer
61 views

How to choose a proper binary operation in a semigroup?

I am interested in generating a finite commutative semigroup which is not a group. And by generating I mean choosing a number $n$ (number of elements in a semigroup) and then defining a $n \times n$ ...
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2answers
293 views

Known bounds for the number of groups of a given order.

The number of nonisomorphic groups of order $n$ is usually called $\nu(n)$. I found a very good survey about the values. $\nu(n)$ is completely known absolutely up to $n=2047$, and for many other ...
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1answer
51 views

Schmidt group and Permute 2-maximal subgroup with 3-maximal subgroup

$G$ is Schmidt group With abelian Sylow subgroup Then every $2$-maximal subgroup of $G$ permuts with all $3$-maximal subgroup of $G$.
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3answers
155 views

A solvable group with order divisible by exactly two primes contains a normal subgroup of prime index.

$G$ is solvable group then $G$ has a normal subgroup $N$ of $G$ such that $|G: N|$ is a prime.
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1answer
41 views

$G$ be a non-nilpotent group and every $2$-maximal subgroup Per with all $3$-maximal subgroup

Let $G$ be a non-nilpotent group. If $|G|=p^{\alpha}q^{\beta}r^{\gamma}$ where $p$,$q$,$r$ are primes (two of them maybe are same) such that $\alpha + \beta +\gamma \leq 3$ then every $2$-maximal ...
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1answer
38 views

Solutions of $ 0 = x^2 -ay^2 -1$ in $\mathbb F_q$ where $a$ is not a square.

Assume $F = \mathbb F_q$ where $q = p^r$ for $p$ prime and $r > 0$. I have to count $$ \{(x,y) \in F^2 \mid x^2 -ay^2 -1 = 0\} $$ where $a$ is not a square in $F^*$. The equation is equivalent to ...
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4answers
549 views

If $(|G|, |H|) > 1$, does it follow that $\operatorname{Aut}(G \times H) \neq \operatorname{Aut}(G) \times \operatorname{Aut}(H)$?

Let $G$ and $H$ be finite groups. If $|G|$ and $|H|$ are coprime, then $$\operatorname{Aut}(G \times H) \cong \operatorname{Aut}(G) \times \operatorname{Aut}(H)$$ holds. What about when $(|G|, |H|) ...
0
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1answer
143 views

Problem 5.15, I. Martin Isaacs' Character Theory

Isaac's Character theory of finite groups book, Problem 5.15: Let $H \subseteq G$ and suppose $\phi$ is a character of $H$ with $det(\phi)=1_{H}$. Let $\chi={\phi}^{G}$ and show ...
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3answers
321 views

Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$

What can we say about the intersection of the $p$-sylow and $q$-sylow subgroups of group $G\;$? It's not necessary that $p=q$. Is there general statements about the intersections of sylows subgroups ...
4
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3answers
330 views

If $H$, $K$ are characteristic subgroups of $G$, when is $\operatorname{Aut}(H\times K) = \operatorname{Aut}(H) \times \operatorname{Aut}(K)$ true?

If $H$, $K$ are characteristic subgroups of $G$, when is $\operatorname{Aut}(H\times K) = \operatorname{Aut}(H) \times \operatorname{Aut}(K)$ true? What if $H$, $K$ are not characteristic subgroups? ...
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1answer
185 views

Questions about cosets, conjugate classes etc

Some questions about subgroups, normal subgroups, conjugate classes etc, just to make sure I understand it :-) The index of a subgroup $H$ in $G$, written as $[G:H]$ is defined as the number of left ...
2
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1answer
119 views

If $G$ is non-nilpotent and $M$ is non-normal subgroup of $G$, then $|G: M|=p^{\alpha}$?

Let $G$ be a finite and non-nilpotent group. Is there a non-normal maximal subgroup $M$ such that $|G:M|$ is a prime power?
6
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2answers
217 views

If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?

If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $? I learnt that if two subgroups are isomorphic then it's not true that they act in the same ...
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1answer
177 views

How to calculate the order of commutator of two elements of a group $G$ in terms of their orders?

if $G$ is a group , $x,y \in G$ and $[x,y]$ is the commutator of $x$ and $y$ so , $[x,y]=x^{-1}y^{-1}xy$ is there a formula to compute $|[x,y]|$ in terms of $|x| , |y|$ ?
2
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1answer
93 views

$G$ is a finite supersoluble group. every maximal subgroup of $G$ permutes with every $2$-maximal subgroup

If $G$ is a finite supersoluble group of order $pq^{\beta}$ such that a Sylow $q$-subgroup $Q=\langle x \rangle$ of $G$ is cyclic and $Q_{G}=\langle x^{q}\rangle$. Then every maximal subgroup of $G$ ...
4
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1answer
92 views

Class of finite groups a Fraïssé Class? [duplicate]

Is the class of finite groups a Fraïssé class? Calling this class $K$, does $K$ satisfy the following: Joint embedding property Amalgamation property Hereditary property: if $G \in K$ and $H \le G$, ...
3
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2answers
164 views

Question about the number of elements of order 2 in $D_n$ [duplicate]

$$\text{Given}\;\; D_n = \{ a^ib^j \mid \text{ order}(a)=n, \text{ order}(b)=2, a^ib = ba^{-i} \}$$ $$\text{ how many elements does $D_n$ contain that have order $2$ ?}$$ My answer would be: We ...
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2answers
48 views

Looking for an integer for which the $(\mathbb{Z}/n\mathbb{Z})^*$ contains elements with certain orders

I don't need a specific answer or whatever, but I'm looking for a strategy to solve this kind of problems. The specific question I have in mind is: Give an integer $n$ for which the multiplicative ...
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1answer
260 views

Homomorphism between multiplicative group of integers modulo n

Just looking for anybody to check the following: We have got a homomorphism $f: (\mathbb{Z}/42\mathbb{Z})^{*} \rightarrow (\mathbb{Z}/21\mathbb{Z})^{*}$, given by $f(a\text{ mod} 42)= a \text{ mod} ...
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0answers
55 views

A family of $8$-regular Ramanujan Cayley Graphs

I'm looking for expander graphs with certain properties. Is there a family of $8$-regular Ramanujan Cayley graphs $\{\text{Cay}(G_n,S_n)\}_n$ such that each of them has no cycles of odd length? ...
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1answer
64 views

$X$-permutable subgroup

Let $A$ and $B$ be subgroup of a group $G$ and $\phi \neq X \subseteq‎ G$. $A$ is $X$-permutable with $B$ if $AB^{x}=B^{x}A$ for some $x$. Let $A, B ,X$ be subgroups of a group $G$ and $K\lhd G$. ...
0
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1answer
323 views

How does one prove U(100) is not cyclic without computing the order of any element?

For starters, $|U(100)| = \varphi(100) = 40$. So by Lagrange, $\forall \hat{x} \in U(100), ord(\hat{x}) \mid 40$. Now, how can one prove there aren't any generators in U(100) without computing the ...
6
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2answers
266 views

What is this method called - Abelization of a Group.

Today, I wanted to make a post for this question. There are some approach in which we can overcome the problem like this and this. According to my knowledge, I could solve the problem via the approach ...
2
votes
1answer
60 views

If $Q\ltimes P$ is a Schmidt group with abelian $P$ and cyclic $Q$, then $\phi(P)$ is trivial.

Let $G$ be a Schmidt group. Suppose that $G=Q\ltimes P$, where $P$ is a Sylow $p$-subgroup of $G$ and $Q=\langle a \rangle$ is a cyclic Sylow $q$-subgroup of $G$. Prove that if $P$ is abelian, ...