# 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.

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### order of an element in a modulo group under multiplication

Suppose $G$ is the group $ℤ_{37}^\times$ under multiplication. Then is there a way that I can prove the order of the element $2$ in $G$ is $36$ without finding all the powers of $2$ until I get unity?...
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### Order of $\mathrm{SL}(n,\mathbb{F}_p)$ (Constructive proof)

Most proofs of $$\vert ~\mathrm{GL}(n,\mathbb{F}_p) ~\vert = \prod_{k=0}^{n-1} (p^n-p^k)$$ I have seen so far, are done by counting the possibilities to build up invertible matrices i.e. counting ...
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### Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
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### $G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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### Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
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### Trying to find an isomorphism

I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ...
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### Automorphism group of finite $p$-groups

firstly I apologize for my naive knowledge in group theory. Let $G$ be a finite $p$-group with automorphism group ${\rm Aut}(G)$ and let $N$ be a maximal subgroup of $G$. Let $g$ be ...
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### Sylow $p$-groups in $GL_3(\Bbb F_p)$

Fix a prime $p$. How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$? Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ ...
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### Can the order of $x \in U_{31}$ be $10$?

Does there exist an element $x\in U_{31}$ such that the order of $x$ is $10$? Here $U_{31}$ is the group of units of $\mathbb{Z}/31\mathbb{Z}$.
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### Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$ [duplicate]

Let $G=\Bbb Z/n \Bbb Z$ and assume $d\mid n$. There exists a unique subgroup of $G$ of order $d$. The mod $n$ reduction of $n/d\in \Bbb Z$ generates a $d$ element subgroup of $\Bbb Z/n\Bbb Z$, ...
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### Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
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### Sigma and Pi Chemistry/Math Permutation Question

Does anyone know if sigma and pi bonds in chemistry have any mathematical definition? The reason I'm asking this is because I've recently read a lot about cycles and permutations, and they seem to ...
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### If $H$ is a subgroup of $G$ of index $n$, then $g^{n!} \in H \ \forall g \in G$

Let $G$ be a finite group and $H$ a subgroup of $G$ of index $n$, i.e., $[G:H]=n$. Prove that $$\forall g \in G,\; g^{n!} \in H.$$ This is a question I've had in a past exam for Group Theory and I'm ...
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### If $H<G$ is abelian, and $\chi(1)=[G:H]$ for irreducible $\chi$, then $H$ contains a nontrivial normal subgroup?

Suppose $\chi$ is an irreducible character of a finite group $G$, and $H$ is a nontrivial abelian subgroup such that $\chi(1)=[G:H]$. Why does $H$ contain a nontrivial normal subgroup? I understand ...
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### 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.
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### $|G| = ?$ 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 ...
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### Trying to show that any group of order four is either cyclic or isomorphic to V

I know the question has already been asked. But I have trouble with the answer. Having a non-cyclic group $\,G=\{1,a,b,c\}\,$, how can I show that $ab=c$? In my attempt, I assume that $ab=1$, and ...
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### Distinct coset representative and stabilizing an element.

Let $G < S_n$ be a permutation group of degree $n$, and let $G^{(i)},0 < i\leq n$, be the pointwise stabilizer of $\{1,2.., i\}$ in $G$. We set $G^{(0)}= G$. For $0 < i\leq n$, let $U_i$ be ...
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### Number of elements of Complete Right Transveral in pointwise stabilizer. [closed]

Let $G < S_n$ be a permutation group of degree $n$, and let $G^{(i)},0 < i\leq n$, be the pointwise stabilizer of $\{1,2.., i\}$ in $G$. We set $G^{(0)}= G$. For $0 < i\leq n$, let $U_i$ be ...
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### Set acting like a Group

I am a little confused with familiar things, so I am looking for some help. Description: Consider the set, $S_3= \{(123), (132), (213), (231), (321), (312)\}$, a symmetric group acting on $3$ ...
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### Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
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### Determine abelian groups with 48 elements

I am just doing some revision for my linear algebra exam, and I came across this problem: Determine all abelian groups (up to isomorphism) with exactly 48 elements. I am not sure I have ever ...
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### For prime $p$, let $G$ be a group such that every non-identity element of $G$ has order $p$. Show that if $|G|$ is finite, then $|G| = p^n$.

I've been self teaching myself some topics in preparation for university and thought I'd have a go at some past paper questions from their website. As such I do not have much experience with these ...
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### Problem with the proof of abelian finite groups decomposition

I can't understand the proof which says that every finite abelian $p$-group can be written as a direct sum of cyclic $p$-groups. I'm using Lang's book of Algebra. My problem is about the following ...
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### Surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$ [closed]

Is there a surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$? I need to find one if the answer is "yes" or explain why the answer is "no". With $\mathbb{Z}_n$ I mean the quotient ...
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### Explicitly calculating the group of global isometries of a convex regular polygon (dihedral group)

Instead of an intuitive geometric description of the dihedral groups $D_{2n}$, that one can find in virtually every good book on group theory, I want to calculate the global isometries of a convex ...
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### Prove /Disprove: Existence of Symmetric Subgroup of Order $n^2-1$ of $S_n$.

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where order of $G$ is $n^2-1$. Problem: Prove (or disporve), Such $G$ exists for $S_n$ for finite $n$. For $n=5,11,71$, ...
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### Can this be done using Sylow theorems? [duplicate]

Let $p$ and $q$ be distinct primes.Suppose that $H$ is a proper subset of the integers and $H$is a group under addition that contains exactly three elements of the set {$p,p+q,pq,p^q,q^p$}.Determine ...
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### Need help in understanding the proof of “If $\vert G \vert$=60 and $G$ has more than one Sylow 5-subgroup, then $G$ is simple.”

This proof is from Dummit & Foote text. Suppose by way of contradiction that $\vert G \vert=60$ and $n_5$>1 but that there exists $H$ a normal subgroup of $G$ with $H$ $\neq$ $1$ or $G$. By Sylow'...
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### Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$. Question: Does $G$ (as defined above for $n=5, 11, 71$) exist? How can I ...
What kind of relation do algebraic equeation and permutation group have? For example, $Z^n -1=0$ is related to a cyclic group $C_n$. Is there anything else in this kind problem? I have read about ...