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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If every Sylow's subgroup is cyclic then $G$ is supersolvable.

I've this exercise to resolve : prove that if $G$ is a finite group and all its Sylow subgroups are cyclic then G is supersoluble. My solution follows: is it correct? Thanks to everyone for the help! ...
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Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove this?...
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On nonabelian $2$-groups of exponent $4$ and center $C_2$.

I am new to this forum, and would like to know whether it is possible to fully classify all nonabelian $2$-groups whose exponent is $4$ and center is $C_2$? Thanks.
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Let $p$ be a Mersenne prime and $G$ an extension of $C_p$, then we have a subgroup in which $C_p$ has a complement

Let $p$ be an Mersenne prime, i.e. $p + 1$ is a power of $2$. Suppose $G$ is an extension of the cyclic group $C_p$ of order $p$ by a group $\overline H$ isomorphic to $PSL(2, p+1)$ or the Suzuki ...
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Combinatorical question related to the number of groups of order $n$

Suppose $n$ is a squarefree number, such that for every prime divisor $p|n$ , there exists at most one prime divisor $q|n$, such that $p|q-1$. If this is the case, then the number of groups of order $...
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What can we say about $D_{2n}$ and $D_n$ if $n$ is even?

We know that if $n$ is an odd natural number then the dihedral group $D_{2n}$ is isomorphic to the group $D_n\times \mathbb{Z}_2$ where $D_m$ is the dihedral group of order $2m$, which has ...
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Equivariant polynomial maps and gradients of invariants

Let $G$ be a finite group with a linear action on $\mathbb{C}^n$ and $f\in\mathbb{C}[X_1,\ldots,X_n]$ be invariant. Then the gradient of $f$ gives rise to a polynomial map $\phi$ from $\mathbb{C}^n$ ...
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The groups with nilpotent hall $p'$ subgroup.

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elemets. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...
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If conjugacy classes and centralizer intersect nontrivially then $G$ is solvable?

Is there any nonsolvable finite group $G$ with the property that $$|x^G\cap C_G(x)|>1 $$ for all $1\neq x \in G$.
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The number of self normalizing subgroup in $G$

Let $G$ be a group define $f(G)$ be the number of proper subgroups with the property $N_G(H)=H$. We can say $G$ is nilpotent if and only if $f(G)=0$. Hence, we can think that when $f(G)$ increase ...
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Given a positive integer $m$, what is the smallest number $n$ such that $S_n$ contains an element of order $m$?

This was a question on a midterm I recently took, but I didn't do well on it (the question I mean), so I'd like to figure out the idea. I thought that since the order of a composition of disjoint ...
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Properties which ${\rm GL}_2(q)$ possess but ${\rm GL}_n(q)$ do not for $n>2$

In every text/reference book on group theory, the commonly introduced family of finite simple groups is $A_n$, and then, some books give a flavor of ${\rm GL}_n(q)$ with conclusion of simplicity of ${\...
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Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ (...
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Relations between the Latin squares of order n and the groups of order n.

Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. ...
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Schur multiplier of “large” groups.

Let $G$ be a finite group and let $M(G)=H^2(G,\mathbb{C}^*)$ be its Schur multiplier. For "small" groups I can compute the Schur multiplier by hand in terms of corresponding roots of unity. However, ...
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How many symmetries does the Cauchy-Schwarz inequality have?

The symmetries of the Cauchy-Schwarz inequality $(x_1^2 + \cdots + x_n^2)(y_1^2 + \cdots + y_n^2) \ge (x_1y_1 + \cdots + x_ny_n)$, as a subgroup of the symmetric group on the $2n$ letters $x_1,\ldots, ...
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Normal subgroup of General linear group

What is the list of all normal subgroups of general linear group $GL_n(q)$? (n*n invertible matrix on finite field with $q$ elements) It is well known $SL_n(q)$ and subgroups of $Z(GL_n(q))$ are ...
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Irreducibility of a certain polynomial associated to an irreducible representation of a finite group

Let $k$ be an agebraically closed field of characteristic 0. Let $G$ be a finite group of order $n$. A representation of $G$ is a homomorphism $\psi: G \rightarrow GL(V)$ where $GL(V)$ is the general ...
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Maximal abelian subgroup of general linear groups

Thanks for any help or comments. Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$. By maximal abelian I mean if $A$ is ...
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How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
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Can $G$ be a union of some of$H$ copies?

Let $G$ be a finite group and $H$ be a proper subgroup. Then we know that $$G\neq \bigcup_{g\in G} H^g$$ I wonder whether this is possible $$G= \bigcup_{\sigma \in Aut(G)} \sigma (H)$$ If there ...
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$\gcd(|G|, |\text{Aut}(G)|)=1$ means G is abelian?

Prove the following assuming that $G$ is finite group with $\gcd(|G|, |\text{Aut}(G)|)=1$. a) G is abelian (done). b) Every Sylow subgroup of $G$ is cyclic of prime order. Since G is ...
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Group of order 112

Let $G$ be a finite group of order $2^4\times 7$ and Sylow $7$-subgroup of $G$ is not normal. Prove that Sylow $2$-subgroup of $G$ is abelian. I am very grateful for any help in this problem.
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a finite group of order $360$ never has an irreducible character of degree $15$

The following problem is an exercise in character theory: "Prove that a finite group of order $360$ never has an irreducible character of degree $15$." Could you help me for it?
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Frattini subgroup of a $p$- group of order $p^4$

Let $p$ be an odd prime and $G$ be a finite non-abelian $p$-group of order $p^4$ with the following presentation: $$\langle a, b, c, d\mid a^p=b^p=c^p=d^p=1, c^d=cb, b^d=ba, [a,d]=[b,c]=[a,c]=[a,b]=1\...
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Galois group of $x^8-2$ and intermediate fields

Let $p(x)=x^8-2$ be a polynomial over $\mathbb{Q}$. I am to find its Galois group $G$ and the correspondence between subfields of its splitting field $\mathbb{K}_p$ and subgroups of $G$. It is easy to ...
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Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
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Groups with more than half elements of order 2

In Dihedral groups, at least half elements are of order two. Question: If a (non-abelian) finite group has at least half elements order two, then what can be said about the group?
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Permutation group of a set

If we let $G$ be a finite permutation group of a finite set $X$ and assume that $G$ has exactly 2 orbits of the same cardinality, how can we show that there is some permutation in G that has no cycles ...
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On transitive constituents of a permuation group

Assume that the intransitive permutation group G has degree n and minimal degree n−1. If no transitive constituent of G has degree 1, then they all are faithful and all except one are regular. I ...
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Sylow $p$-subgroup of $GL_{n}(\mathbb{F}_{q})$

This question has already been asked here. However, the answer there isn't particularly helpful. Consider $G = GL_{n}(\mathbb{F}_{q})$ where $q = p^{r}$ where $p$ is prime. Show that the upper ...
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About motivation of Lang's Proof $S_n$ is not solvable for $n\geq 5$.

In Lang, he proves that $S_n$ is not solvable if $n\geq 5$ by using following observation. If $N\unlhd H\leq G$, H contains every 3-cycle, and if $H/N$ is abelian, then H contains every 3-cycle. Where ...
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Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
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Abelian group action exercise

Let $X$ be a set with $n$ elements and let $G$ be an abelian group acting on $X$ such that: $$(i) \space gx=x \space \forall x \implies g=1,$$ $$(ii) \space \forall x,y \in X, \exists g: gx=y.$$ Show ...
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If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian. What I am trying that G/F is ...
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How can we find element orders of a finite group?

Suppose $p$ and $q$ are two prime divisor of the order of a finite group $G$. I want to know if $G$ has an element of order $pq$ using the character table of $G$. Is this possible? If so, please ...
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Simple question on Sylow subgroups.

Given a finite group $G$, $|G|=p^am$, $P,Q\in\operatorname{Syl}_p(G)$, then we know that $Q=P^x$ for some $x\in G$. Thus every element of $Q$ is the image of an element of $P$ under the isomorphism ...
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Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
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Constructing groups from actions

As an example to set the scene, suppose I have a cube and I let its symmetry group act on its faces. Every face admits four transformations which leave the cube invariant and which maps the chosen ...
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Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq i}^n\frac{\sin(\theta_j-\theta_i)}{\left(1-\cos(\theta_j-\theta_i)\right)...
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What is the difference between operator groups and group actions?

I have always thought that operator groups and group actions are two names for the same thing. Now I have noticed that they have different codomains. Actually I am still confused, why are they ...
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Central extensions of elementary abelian p-groups

Given an elementary abelian $p$-group $E$, we can consider $E$ as a trivial $E$-module; my first question is : How can one compute the rank of of the cohomology group $\operatorname{H}^n(E,E)$, $n \...
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On automorphisms of finite abelian group

Let $G$ be a finite abelian group such that $a, b\in G$ and $\mid a\mid=\mid b\mid$. Then does there exist an automorphism of $G$ such that $\alpha(a)=(b)$? Thank you
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Groups of order 8

Ok so I am looking at a proof to all the groups of order 8, I've attached an image which basically is the start of the proof. In particular the part highlighted with yellow is causing me problem, I ...
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Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
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Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a n-...
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An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
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Action of groups on graphs

Let $K$ be a simplicial graph [i.e. no loops and no double edges]. Let $G$ be a group which acts on $K$ by graph isomorphisms. Suppose that the action satisfies: $K$ is connected finite number of ...