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Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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Structure of сonjugacy subgroup intersection graph

Let $G$ be a finite group, $H$ is proper subgroup and ${\cal H}(H)$ the set of all subgroups of conjugate H. Construct the graph $\Gamma$, with vertices ${\cal H}(H)$ and two subgroups adjacent if ...
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Maximality on non-nearly normal subgroups and derived subgroup

I have troubles understanding this lemma I found in a paper on maximality of non nearly normal subgroups. Statement: Let $G$ be a group satisfying $Max-nn^{-}$. Then the commutator subgroup $G'$ of $...
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1answer
34 views

Lower bounds on the number of subgroups of a specific group

I am visiting this thread: $G$ is Abelian iff all the Sylow subgroups of $G$ are normal I am wondering whether more can be said about the lower bounds of number of subgroups of $G$ when $G$ is of ...
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38 views

Cayleys Theorem worked example

Find the smallest n such that $S_4 \times S_8$ is isomorphic to a subgroup in $S_n$. Answer By Lagrange, we know that the minimum $n$ has to be at least $12$. But also if $n=12$ then this is easy to ...
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A special case of group extensions, other then the semidrect product.

It's a well-known fact that if $A$ an Abelian group and $G$ is a group, then all group extension of $G$ by $A$ is isomorphic with the group ($A\times G,\,\bullet)$, where the group operation $\bullet$ ...
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1answer
20 views

A relation between FC groups and locally finite groups

It is proved that if $G$ satisfies maximality on non-nearly normal subgroups, denoted with $Max-nn^{-}$ and $K$ is a cyclic subgroup of $G$, then $K^{G}$ satisfies the maximal condition on subgroups. ...
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1answer
32 views

Centralizer of Sylow $p$-subgroups for nonabelian finite simple groups

Given a nonabelian finite simple group $G$, is it always true that $$C_G(P)\leq P$$ for each Sylow $p$-subgroup $P\leq G$?
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1answer
43 views

The Alternating Group $A_n$

We know that $\forall n\geq 5,$ the alternating group $A_n$ is simple. But $n\leq 4,$ Is the alternating group $A_n$ simple? I can't find examples of them . Please help me! Thank you.
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FC-groups with maximality on non-nearly normal subgroups

I am reading this proof about characterization of FC-groups having maximality on non-nearly normal subgroups, which I will denote with Max-nn$^{-}$. Recall that $H$ is a nearly normal subgroup of a ...
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0answers
35 views

What is the name of the group?

What is the name of the subgroup of the affine plane group generated by the transformations of the form $$ x \mapsto a \begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta &\cos\theta \end{...
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1answer
30 views

Normal subgroup but not characteristic subgroup

I know every characteristic subgroup is normal subgroup but converse is not true. I find an example in Klien 4- group. But I have seen on internet that in Q$_8$, each of cyclic subgroup of order 4 ...
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3answers
34 views

Example of proper image of center of group under epimorphism.

Let $f:G \to H$ be a group epimorphism and let $Z(G)$ be a center of a group $G$. We already know that $f(Z(G)) \subseteq Z(H)$. Is there any small example showing $f(Z(G))$ is a proper subgroup of $Z(...
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1answer
59 views

Let $G$ be a group of order $8$ and $y$ be an element of $G$ of order $4$. Prove that $y^2 \in Z(G)$ [duplicate]

The question Let $G$ be a group of order $8$ and $x$ be an element of $G$ of order $4$. Prove that $x^2 \in Z(G)$ already posted here. But the answer is not given there. SO I have tried to solve the ...
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1answer
64 views

Center of a group with order $p^aq^b$

Are there any examples of groups, $G$, such that $|G|=p^aq^b$ (where $p$ and $q$ are distinct primes and $a,b\geq 1$) and $|\mathrm{Z}(G)|=p^a$? ($\mathrm{Z}(G)$ denotes the center of $G$) I ...
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1answer
51 views

Universal Property for Presentations

Let $G$ be a group with a presentation $\langle S | R \rangle $ ( $G = \langle S|R\rangle$), with an associated map $f\, : S \rightarrow G$ Let $\bar{N}_{R}$ be the normal closure of $R$ in $F_{S}$. ...
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1answer
52 views

Is there a cyclic subgroup of $A_8$ of order $10$

I'm trying to check if there is a cyclic subgroup of $A_8$ of order $10$. I believe that it does not have one. I think that we can prove it by showing that it can't get build. The problem is more ...
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1answer
46 views

Product, homomorphism and kernel of these two additive matrices groups [closed]

Consider the additive groups $\mathit{G}:= \mathfrak{M}_{\mathrm{4x1}}(\mathbb{Z_{11}})$ and $\mathit{H}:= \mathfrak{M}_{\mathrm{3x1}}(\mathbb{Z_{11}})$ (column matrices with height 4 and 3 ...
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0answers
33 views

Are there nontrivial subsets of the complex unit circle satisfying the multiplicative Jacobi identity?

Let $a^{(b^c)}\times b^{(c^a)}\times c^{(a^b)}=1$ Then a set $S$ satisfies the multiplicative Jacobi identity if this is true for all $a,b,c\in S$. $S=\{1\}$ satisfies the identity. $S=\{0\}$ doesn'...
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2answers
38 views

Union of two characteristic subgroups

I have proved intersection of two characteristic subgroup is characteristic but when I want to prove this for union I got stuck. Is union of two characteristic subgroup is characteristic subgroup? ...
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1answer
49 views

The fundamental group of the gluing of two genus $g$ $3$-dimensional handlebodies

I was given the problem above, and I'd appreciate some help. I think I have a general direction, but I'm not entirely sure if what I'm doing is true, so it'd be great if someone could tell me if what ...
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3answers
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Finding $\tau\in S_9$ for $\tau(1,2)(3,4)\tau^{-1}=(5,6)(1,3)$

I'm trying to understand where I'm wrong in my solution. I would like to find all $\tau \in S_9$ so $$\tau(1,2)(3,4)\tau^{-1}=(5,6)(1,3)$$ Meaning - $(\tau(1),\tau(2))(\tau(3),\tau(4))=(5,6)(1,3)$. ...
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1answer
47 views

If G has a unique subgroup H of a given (finite) Index, then Prove that H is characteristic subgroup of G

I know if a group G has a unique subgroup H of a given order then H is characteristic subgroup of G as every automorphism f on G order f(H) = order H . But here H is unique subgroup of G of given ...
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1answer
61 views

Explicit description of the conjugation action of $[[1,0],[0,p]]$ on the amalgam $SL(2,\mathbb{Z})*_{\Gamma_0(p)} a_pSL(2,\mathbb{Z})a_p^{-1}$

Let $a_p$ denote the matrix $[[1,0],[0,p]]$, where $p$ is prime. Then $SL_2(\mathbb{Z}[1/p])$ can be presented as the amalgamated product $$SL_2(\mathbb{Z})*_{\Gamma_0(p)} a_pSL_2(\mathbb{Z})a_p^{-1}...
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1answer
56 views

Finding coproducts in $\mathbf{Grp}$ using presentations

Aluffi II.8.7 suggests proving that, given groups $G, G'$ admitting representations $(A \mid R), (A' \mid R')$ respectively, where $A, A'$ are disjoint, their coproduct in is $(A \cup A' \mid R \cup R'...
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1answer
34 views

Is addition on a circle equivalent to addition of integers [closed]

If you do modulo arithmetic on a circle you have a certain group. But if you fourier transform a function on a circle you get discrete values. (Compared with continuous fourier tranform of a function ...
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2answers
49 views

Klein four groups, cyclic groups

A basic definition question: "the Klein four group V is the simplest group that is not cyclic." Does this simply mean you need two (non-identity) elements of the group to generate the entire group? ...
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1answer
43 views

Visualization of the relationship between Conjugacy Classes and Centraliser

I've recently come across a theorem in my abstract algebra course which states that for a finite group $G$ and for any $a\in G$ $|\text{cl}(a)|=[G:C(a)]$ where $\text{cl}(a)$ is the conjugacy class ...
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3answers
501 views

Can all groups be thought of as the symmetries of a geometrical object?

It is often said that we can think of groups as the symmetries of some mathematical object. Usual examples involve geometrical objects, for instance we can think of $\mathbb{S}_3$ as the collection of ...
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1answer
76 views

Uniqueness of Rank of Free Group via Abstract Nonsense

It is a well known result that given two isomorphic free groups, their freely generating sets have the same cardinality, which is then called the rank of a free group. I am well aware of the usual ...
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0answers
18 views

Modulo-density of binary numbers with fixed digits

Let $var \subseteq \{1,...,n \}$. Let $a=|var|$. Let $p > 2$ be a prime number. Let $0 \leq q < p$ be a fixed remainder modulo $p$. $$M = \{w \in \{0,1\}^a : \text{number}(\widehat{w}) \equiv q ...
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4answers
62 views

General Linear Group: Thinking of invertible matrices as bijective functions?

In my lecture notes I encountered the following result, For any set $S$ the set $F$ of bijective functions $f:S\to S$ is a group under composition, but is not in general abelian. Then it was ...
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50 views

Finite group with subgroup containing all its centralizers

Let $G$ be a finite group. $H \leq G$. Suppose $\forall x \in H\backslash\{1\}, C_G(x) \subset H$. Prove that gcd$(|H|, [G:H]) = 1$. This was a homework question from last semester that I'm trying to ...
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1answer
33 views

Number of subgroups of index 2 in $(C_2)^3\times C_3$

I need to count the number of subgroups of index 2 in $(C_2)^3\times C_3$. I think there are 3, because we have to take $C_3$ and then we have 3 choices for a $C_2$ not to pick. However, a classmate ...
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0answers
34 views

Reference Request: An operation preserving bijection from a magma to a group must be a group isomorphism

Let $M$ be a magma with a binary operation $*_M$ and let $G$ be a group with a binary operation $*_G$. If $f$ is a bijection from $M$ to $G$ preserving the operation, that is, $f(m_1 *_M m_2)=f(m_1)...
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2answers
49 views

Prove that 2 cyclic subgroups of same order are either equal or contain $e$ only as intersection? [closed]

Can someone give me a proof that $2$ cyclic subgroups of same order in a group are either equal or their intersection is $e$. I think Lagrange's theorem is required for it. Also suggest some visual ...
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1answer
84 views

Every finite group is finitely presented, but how to do this effectively?

Aluffi II.8.3 suggests proving that every finite group can be finitely presented. Clearly, we could just present a group via its whole underlying set as the set over which we construct the free group,...
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1answer
71 views

Proving that every group of order 48 is not simple

I know that this question has been asked a lot, but I´m trying with a different approach(I think). The proof it´s divided in two parts: Let G be a simple group of order 48. 1)If $S_1,S_2,S_3$ are ...
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1answer
20 views

Does regular action preserve cyclic subgroup generation?

If you have three elements $x, y, z \in G$, it isn't hard to show that if $\langle y,z \rangle$ is cyclic, then $\langle x^{-1}yx, x^{-1}zx \rangle$ is cyclic. However, is it also true that $\langle ...
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1answer
41 views

Action of Adjoint Map on Lie Algebra

My question is specifically concerning the Lie algebra $su(N)$. Since this is a compact, real and semi-simple Lie algebra, for each element $X\in su(N)$ there exist some $A,B\in su(N)$ such that $$X=[...
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2answers
80 views

Every subgroup of cyclic group is cyclic,How do I visualize this fact graphically? [closed]

I want to visualize everything in Mathematics.Is there a way to visualize the theorem I stated above?Can it be represented graphically?I want to be very analytical in this topic.So please can anyone ...
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0answers
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+50

Does $\Sigma$ generate the variety of all groups?

Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
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2answers
242 views

Can the concepts of abstract algebra be visualized as in analysis?

I like to visualize everything I study but yet I have found pretty nothing to visualize in abstract algebra.I have studied group theory upto subgroups Cyclic groups and Cosets and Lagrange's theorem....
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1answer
52 views

How can I visualize an infinite cyclic group.I know that it looks like $(\mathbb{Z},+)$ [closed]

I want to visualize infinite cyclic group. I know it looks like $(\mathbb{Z},+)$.But actually I want to visualize it as a limiting case of finite cyclic group when the order of group gradually gets ...
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0answers
39 views

There is a formula for polynomial equations if and only if the generic polynomial is solvable

I'm trying to understand the proof that if $F$ is any field of characteristic $0$ there is no general formula for solving polynomial equations of degree $5$ or higher. I know the following facts: $1. ...
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1answer
94 views

On finite simple groups whose orders are perfect powers

A short note in Group Atlas v2.0 states: $\mathrm{PSp}(4,7)$ is the smallest simple group whose order is a proper power. Question: Is there other known finite simple groups whose orders are ...
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1answer
73 views

how many abelian transitive subgroups of $S_{n}$

It's well-known that any abelian transitive subgroup A of a symmetric group $S_{n}$ has order $n$. Moreover, does anyone know how many abelian transitive subgroups of $S_{n}$ and what do they look ...
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29 views

Finding the image of an element under an automorphism by GAP

I am doing GAP to find the image of an element under an automorphism of a certain group: ...
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0answers
32 views

Irreducible representations of $\mathbb{C}^{*}$

Suppose you have a smooth irreducible representation (finite dimensional over $\mathbb{C}$) of $G = \mathbb{C}^{*}$, then does it have to be of the form $z.w = z^n w$ for some $n \in \mathbb{Z}$ and $...
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1answer
22 views

Minimal generating set of octahedral/cube group

I am aware that $S_4\times Z_2$ is the full symmetry group of the octahedron and the cube (since they are duals). I've found a relatively straightforward way to generate this group with $3$ elements, ...
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2answers
145 views

Why is a + a = 2a in a group?

Suppose we have a group A, with an element a. We write that a + a = 2a. This doesn't mean that we've introduced a multiplication operation $*: \mathbb{Z} \times A \rightarrow A$, as far as I can tell. ...