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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Size of maximal tori in finite simple groups of Lie type

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$ an odd prime. Is it possible that $G$ contains a maximal torus of order $2^m$ for some positive integer $m$? ...
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Suppose that $|G| = p^aq$, where p and q are primes and a > 0, Then $G$ is not simple?

Proof : We can assume that p$ \neq $ q and $n_p >1 $, so $n_p$ = q. Now choose distinct Sylow p- subgroups $S$ and $T$ of $G$ $S\cap T|$ is as larger as possibe and write $D = S \cap T$. if $D$ ...
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Ruling out orders when applying Sylow's theorems

Going through examples of applications of the Sylow theorems in Fraleigh's book, when proving that no group of order 36 is simple, after concluding that $| H \cap K | = 3$ for two $3$-Sylows $H$,$K$, ...
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Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$.

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$. Also another question, do p and q have to be distinct for this to hold? On top of ...
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Dihedralize Twice - dihedralize a dihedral group $D_n$

A simplest dihedral group $$D_4=C_2 \ltimes C_4$$ can be regarded as dihedralizing a $C_4$ by a semi-direct product. Q: Can one dihedralize the group $D_4$ a second time by defining $$C_2 \ltimes ...
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Show that if the prime $p$ divides $|G|$, then $|X|$ is divisible by $p$.

Question : Let $p$ be a prime number that divides the order of the finite group $G$. Let $X$ = $\bigcup_{P \in Syl_p(G)}P$. Show that $|X|$ is divisible by $p$.
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Expressing a matrix in terms of subgroup generators using Magma

With Magma, it is possible to define a subgroup $H$ of a finite matrix group $G$ in terms of generators. Given a matrix $M\in G$, Magma can also determine whether $M\in H$. Presumably, if Magma ...
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Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
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Intersection subgroup in cyclic group [on hold]

Let $G$ be a finite group and $x$ is a nontrivial element of $G$. If for every nontirivial element $y\in G$, $\langle x\rangle\cap\langle y\rangle\neq 1$, then is $G$ cyclic or generalized quaternion ...
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Intersection subgroup in cyclic group [duplicate]

Let $G$ be a finite group and $x$ a non-trivial element of $G$. If, for every non-trivial element $y\in G$, $\langle x \rangle \cap \langle y \rangle \neq \{1\}$, then is $G$ cyclic or generalized ...
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Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$

Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$ Attempt: $Z_p \oplus Z_p$ has $p^2-1$ elements of order $p$ . Hence, all non trivial elements of $Z_p \oplus ...
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1answer
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Online Finite Field Calculator

I need to find an online Finite Field calculator with the following operations: Inverse SqrRoot Mult Div I have found one a couple of days ago but lost the url, and cannot find it now. Any ...
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75 views

If G acts on X, show that there must be a fixed point for this action. Please help. [on hold]

Suppose that G is a group of order p^k, where p is prime and k is a positive integer. Suppose that X is a finite set and assume that p does not divide the size |X| of X. If G acts on X, show that ...
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Let G be a group and let diag G denote the subset of GxG defined by diag G:={(g,g) |g∈G}(The subset diag G is known as the diagonal of G in GxG.) [on hold]

Let G be a group and let diag G denote the subset of GxG defined by diag G:={(g,g) |g∈G}(The subset diag G is known as the diagonal of G in GxG.) (a) Show that diag G is a subgroup of GxG. (b) Show ...
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Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
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38 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
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A Criterion for being Sylow p-group

Show that if $H$ is a $p$-group of finite group $G$ and $N_G(H)=H$ then $H$ is a Sylow $p$-group of $G$? Or prove the following more general property,$$[G:H]\equiv1\ (\mod\ p)$$
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Groups with single conjugacy class of subgroups. (modified)

I am going to modify my previous question. What are those finite non abelian groups in which non normal subgroups of same order are conjugate. e.g. Dihedral groups of order $4n+2$.
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Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
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1answer
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Visualizing the 48 actions of GL(2,3)

Hello and thank you for your patience. (DISCLAIMER: I'm a novice and not a mathematician by trade and I'm not certain how to articulate most of my questions here. I am learning from experiences and ...
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What is the most elementary proof of the following: a non-abelian group of order $6$ is isomorphic to $S_3$

We know that, a non-abelian group of order $6$ is isomorphic to $S_3$. While I've been able to locate different proofs of this result, I would like to have one that is as elementary as it can be. So ...
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1answer
38 views

An assumption used in defining a group of order $mn$

I want to show that the defining relations $a^m=b^n=e, ab=ba^k$ define a group of order $mn$ with a normal subgroup of order $m$, if $k^n \equiv 1 \pmod m$. Consider the set of symbols of the form ...
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How do I effectively solve this kinds of problems?

I'm preparing for a test on Monday. This is an exercise in the past homework. Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$. Is there a ...
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1answer
60 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
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50 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
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The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
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To Find a subgroup of order $6$ in $U(700).$

Find a subgroup of order $6$ in $U(700).$ Attempt: $U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$ $\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$ ...
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1answer
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Is this type of a subgroup always normal? [duplicate]

Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or ...
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1answer
51 views

If G is a finite group and $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$

Let G be a finite group. Show that, given $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$. I'm trying to use the info that is finite, but I can't find a way. For instance, if G is a ...
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What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
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An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
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An algorithm for generating a finite group with a finite set of generators

Let $A$ be a finite set of permutations on $\Bbb N$ with finite support. Is there a good efficient algorithm to obtain the subgroup $\left<A\right>$ of the symmetric group ...
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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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Groups - Inversions

Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the ...
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conjugacy classes and order of group

Suppost that $k_G(A)$ denotes the number of conjugacy classes of $G$ that intersects $A$ non-trivially ($A$ is an arbitrary subset of $G$) and $M=G^{'}Z(G)$. Also suppose that $G$ is non-solvable, ...
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Conjugacy classes of right cosets

Is it true that all elements of a right coset $Hx$, for a subgroup $H$ of $G$, contained in a unique $G$-conjugacy class? I mean if $Hx=\lbrace{x_1,...,x_s}\rbrace$, then is it true that ...
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permutation problem: cycle representation

Let $n$ be an odd number. Let $C_n$ be the set of permutations $\pi$ of $[n]$ whose cycle representation has only one cycle. Let $\pi,\sigma\in C_n$. Prove that their composition $\pi\sigma$ has an ...
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1answer
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Questions on proof that transvections are conjugate in $GL(V)$.

I have difficulty following the proof that transvections are conjugates in $GL(V)$, and for $n \ge 3$ even in $SL(V)$. I give the necessary definitions and the proof, with the problematic parts ...
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Groups with single conjugacy class of subgroups. [duplicate]

I wish to know all those groups in which there is single conjugacy class of subgroups of fixed order.For example, In finite cyclic groups and in Alternating group of degree 4, number of conjugacy ...
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Find all sub groups of order $4$ in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$ . Are they all cyclic?

Find all sub groups of order 4 in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$. Solution : $\mathbf{Z}_4 =\{0,1,2,3\}$ $O(1) = O(3) = 4$, $O(0) = 1$, $O(2) = 2$ Hence, I found the subgroups of order 4 as ...
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$|G| \geq 3 $ is finite and has subgroup with index more than 4 then $G$ is not simple.

suppose $G$ is a finite group with more than three elements,prove that if $H$ will be a proper subgroup of $G$ with this property that $[G:H] \geq 4$ then $G$ is not simple. I found this question a ...
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Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
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1answer
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Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
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If $G$ is solvable and $G/[G,G]$ is cyclic, can $G\times G$ be generated by 2 elements?

I doubt this is true, but I haven't found any small counterexamples (there are no counterexamples with $|G| < 1536, |G| \neq 768$): Suppose $G$ is finite, solvable, 2-generated, and $G/[G,G]$ ...
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Normal Sylow $p$-subgroup of a normal subgroup

Any hints for the following question - I am sure that I am missing something very simple here. $K$ is a normal subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $K$. If $P$ is a normal subgroup of ...
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Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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1answer
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Splitting field of an irreducible polynomial of degree five $f \in \mathbb{Q}[X]$

Let $\Omega/\mathbb{Q}$ be the splitting field of an irreducible polynomial $f \in Q[X]$ of degree five. Show that $Gal(\Omega/\mathbb{Q})$ equals $A_5$ or $S_5$ if $Gal(\Omega/\mathbb{Q})$ has an ...
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Group Theory (Abstract Algebra) question! o(G) vs o(g)??

If G is a group, and g is an element of G, what is the difference between the following two notations o(G) and o(g)?
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How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...