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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Proving $G\simeq G^{op}$

Let $G$ be a group, regarded as a one-object category all of whose maps are isomorphisms. Then its opposite $G^{op}$ is also a one-object category all of whose maps are isomorphisms, and can ...
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1answer
38 views

A strange result if $|G/\mathrm{Z}(G)|=p$

I came across something strange, which I would like to share. Let's take a group $G$ such that $|G/\mathrm{Z}(G)|=p$, where $p$ is a prime number. Then, we can show that $G$ is abelian $\iff \...
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1answer
20 views

When regarding groups as categories, how do I know whether $g\cdot h=g\circ h$ or $g\cdot h=h\circ g$?

A group can be regarded as a category with one object in which all arrows are isomorphisms. As a set, the group corresponds to the set of morphisms, and the group operation corresponds to the ...
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1answer
24 views

Find all Quotient (or Factor) Groups of D4 (Dihedral Group 4)

I need to be able to find all of the quotient groups for dihedral group 4 with $D_4=${$e,R,R^2,R^3,V,H,D,D'$}. I know I have to start by finding the normal subgroups, which are {$e,R^2$} {$e,R,R^2,...
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1answer
48 views

Subgroup of order $10$ must be normal?

Let $G$ be a group such that $H \le G $ and $o(H) = 10$. Does there exist a group $G$ with a subgroup $H$ such that $H$ is not normal subgroup of $G$? my intuition is that there is no such a group $...
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1answer
40 views

Additive group of $\mathbb{Z}[x]$, ring of polynomials over integers. [duplicate]

Let $\mathbb{Z}[x]$ be the ring of polynomials over integers. Then what will be the additive group $\mathbb{Z}[x]$ ??. Which is/are true?? 1) It is isomorphic to set of rational numbers under ...
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If $\phi(fg)=\phi(f)g$, how to call it? [on hold]

Given $f$ and $g$ belongs to group $G$ and $\phi$ a group action of $G$, if $$\phi(fg)=\phi(f)g$$ , how to call this property ?
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2answers
40 views

What subgroup of $S_4$ is this?

What is the subgroup of $S_4$ generated by $\left(14\right)\left(23\right)$ and $\left(12\right)$? I can see that they are both order 2, and they don't commute, but can't see where to go from here.
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24 views

parametrization for an explicit matrix group

Let $G$ be the subgroup of $(P)SL(2,\mathbb{C})$ generated by $\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array} \right) $ and $\left( \begin{array}{cc} 1 & 0\\ -j & 1 \end{array} \...
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57 views

When the sum of two permutations is a permutation? [on hold]

I have to work with encrypted data, a problem arises as follows: Assume that we have $\sigma_1, \sigma_2 \in S_X$, where we consider the set $X$ as the abelian group $Z_n$. Each element of $X$ is an ...
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4answers
47 views

Subgroup of order 2 of a group of order 56 [duplicate]

I was given the following question: Does every group of order 56 contain a subgroup of order 2. I know that the Sylow theorems guarantee the existence of an order 8 subgroup. Is there a general ...
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1answer
33 views

Suppose $AN_1=AN_2=G$, $A\cap N_1=A\cap N_2=1$ and $N_1,N_2$ are normal in $G$, do we have $N_1\cong N_2$?

Let $G$ be a finite group. Suppose $AN_1=AN_2=G$, $A$ is a subgroup of $G$, $A\cap N_1=A\cap N_2=1$ and $N_1,N_2$ are normal in $G$, do we have $N_1\cong N_2$? I think this is not true,but I failed ...
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30 views

A condition for normalized 2-cocycles from the existence of the inverse element in a group extension

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

A functor $\mathcal G\to \mathbf{Set}$ is the same as a left $G$-set

I'm trying to understand the first part of Example 1.2.8 from here: https://arxiv.org/pdf/1612.09375.pdf Let $Ob(\mathcal G)=\{\star\}$. A functor $F:\mathcal G\to \mathbf{Set}$ consists of: An ...
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0answers
3 views

Uniform powerful pro-$p$ group isomorphic to $Z_{p}^{d}$

Problem. Let $G$ be a uniform pro-$p$ group and suppose that $G$ has an abelian open normal subgroups. Show that $G/Z(G)$ is finite, deduce that in fact $G \simeq \mathbb{Z}_{p}^{d}$ for some $d$. (...
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3answers
92 views

Is $\mathbb{Z}_{4} \oplus \mathbb{Z}_{6} \cong \mathbb{Z}_{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4}$?

Is $\mathbb{Z}_{4} \oplus \mathbb{Z}_{6} \cong \mathbb{Z}_{2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{4} $? In my opinion, this statement is correct because the maximal order of element in each ...
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0answers
33 views

Is there a group $G$ such that $\mathrm{Aut}(G)\simeq (\mathbb{Q},+)$? [duplicate]

My guess is that no such $G$ exists. All I know is that if $\mathrm{Aut}(G)$ is abelian, then $G$ is two-step nilpotent. Since $\mathbb{Q}$ is locally cyclic (i.e. every f.g. subgroup is cyclic), can ...
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43 views

Free Groups and Actions on Trees

Here is the proof of a theorem I am working through in Geometric Group Theory by Clara Loh: The first paragraph shows that if $F$ is free, then it admits a free action on a (non-empty) tree. This ...
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23 views

Finding basis for a representation of $D_8$.

Let $G=D_8=\langle a,b\mid a^4=b^2=1,b^{-1}ab=a^{-1}\rangle$. The character table of $D_8$ is known and is Let $U:=\bigg\{\sum\limits_{1\leq i<j\leq 4} a_{ij}x_ix_j\mid a_{ij}\in\mathbb{C}\bigg\}$ ...
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1answer
52 views

Every endomorphism $\psi$ of $\pi_1(S^1\times S^1, (1,1))$ can be expressed as $\psi = f_\ast$?

I have to prove the following: Using the canonical isomorphism $$\pi_1(S^1\times S^1,(1,1)) \cong \pi(S^1,1)\times \pi_1(S^1,1),$$ show that every endomorphism of the group $\pi_1(S^1\times S^1,(1,...
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2answers
143 views

On group quotient by a characteristic subgroup

My question is: Let $H,K\leq G$ be two characteristic subgroups and assume $H\leq K$. Do we have $K/H$ is characteristic in $G/H$? We know that any characteristic subgroup of $G/H$ must be of the ...
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1answer
30 views

Value of Irreducible Character in Quotient Algebra

Let $G$ be a finite group an $F(G)$ the algebra of functions on $G$. Let $N\lhd G$ be a normal subgroup and consider the ideal: $$J_N=\{f\in F(G)\,|\,\forall\,n\in N,\,f(n)=0\}.$$ Consider the ...
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1answer
68 views

How to show that a group homomorphism from $SL(2, \Bbb Z/n\Bbb Z)$ is one-to-one?

Let $n$ be an odd integer, and let $f:SL(2, \mathbb{Z}/n\mathbb{Z}) \to GL(n^2, \mathbb{C})$ be a group homomorphism. The special linear group $SL(2, \mathbb{Z}/n\mathbb{Z})$ is known to have the ...
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2answers
84 views

Disproving the converse of Lagrange's theorem

In this page of wikipedia there is a disproving of the converse of Lagrange's theorem. I would like to see a more simple (or short) disproving of Lagrange's theorem.
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12 views

Name for an algebra formed by the complex field extension of a Lie group?

If we take a lie group for example $SO(3)$ which is a 3-dimensional group with an infinite number of elements of the group. Then we take the complex field extension of this $\mathbb{C}[\{g|g\subset SO(...
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23 views

Finite Sylow subgroups of periodic groups

Let $G$ be a (possibly infinite) periodic group, and suppose that $G$ admits a maximal finite $p$-subgroup $P$. By this I mean that we do not assume that $P$ is not strictly contained in any other $p$-...
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45 views

What does 'every monomorphism in $R$-$\mathsf{Mod}$ is a kernel' mean?

In Algebra: Chapter $0$ by Aluffi, at the end of page 161, he writes: Since every submodule $N$ is then the kernel of the canonical projection $M \to M/N$, our recurring slogan becomes, in the ...
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1answer
30 views

Group with minimal order which maps to a given subgroup under homomorphism

Let $\psi:{H}\mapsto{H}'$ be a surjective homomorphism. Can there exist a proper subgroup ${M}$ of ${H}$ such that $\psi({M})={H}'$? Can there exist more than one such subgroup? Should they be ...
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2answers
36 views

How to show that there exists an order $p$ normal subgroup if $|G|=4p$ with $p\geq 5$ prime.

Let $G$ be a non-abelian group of order $4p$ with $p\geq 5$ prime. Then there exists a normal subgroup $H\leq G$ of order $p$. I have proven that $G$ has four one-dimensional irreducible ...
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20 views

$J= \{g\in G: ||g||_S<n\}$ for some $n\in\mathbb{N}$ is a finite set?

Let $G$ be a finitely generated group and let $S$ be a finite generating set (For convenience, we always assume our generating set is symmetric, i.e. $s\in S$ implies that $s^{-1}\in S$). We define ...
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1answer
33 views

Showing sets of the same cardinality have isomorphic symmetry groups

Let $X$ and $Y$ be sets s.t. $|X|=|Y|$. Show that $Sym(X)\simeq{Sym(Y)}.$ By the first assumption there exists a bijection $\phi{:Sym(X)\rightarrow{Sym(Y)}}.$ I remain to show that this is a ...
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0answers
21 views

Show that don't exist a linear representation for the Galilei's group.

Let G Galilei's group and R a projective representation, show that don't exist $\phi : G\longrightarrow \mathbb(C) $ such that $V: G \longrightarrow U(H) $, $V(g)=\phi(g) R(g) $ such that $V(g_{1}g_{2}...
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1answer
35 views

Show that every Frobenius group contains a solvable Frobenius subgroup.

I have trouble in the problem ''Show that every Frobenius group contains a solvable Frobenius subgroup." My idea is to take a subgroup of order $p$ in the Frobenius complement and show there is a ...
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Show that $HA_n/A_n$ is cyclic of order $2$ [on hold]

Let $n\geq 5$, $H=\{(1 \ 2), \ (3 \ 4 \ 5)\}\leq S_n$ and $A_n$ is the subgroup of even permutations. Show that $HA_n/A_n$ is cyclic of order $2$. For that do we have to calculate first $HA_n$ ?
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1answer
52 views

Show property of a group element of infinite order

Let $G$ be a group and $a\in G$ of infinite order such that $\langle a\rangle \leq G$ (normal subgroup). Show that $g^2a=ag^2$ for each $g\in G$. Could you give me a hint how we could show that? ...
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0answers
44 views

Number of subgroups of order $p^k$ in $p$-groups

Let $G$ be a non-abelian $p$-group and $p^k\ge p^3$ be a proper divisor of $|G|$. It is well known that the number of subgroups of order $p^k$ in $G$ is $1\pmod{p}$. Divide the subgroups of order $...
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2answers
55 views

Let $P$ be a Sylow $5$-subgroup of the symmetric group $S_{25}$ on $25$ letters. Show that there are two elements of $P$ that generate $P$ as a group.

Let $P$ be a Sylow $5$-subgroup of the symmetric group $S_{25}$ on $25$ letters. Show that there are two elements of $P$ that generate $P$ as a group. We know that the order of $P$ is $5^{\left\...
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0answers
39 views

Inverse Limit of Finite Direct Product of Groups/TopSpaces

I’m wondering if the inverse limit of a finite direct product of groups $G_{i,j}$ or topological spaces $X_{i,j}$ is isomorphic to the direct product of the inverse limits of $G_{i,j}$ or $X_{i,j}$, ...
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1answer
33 views

Formula of a the group of character of a group

If I have a finite abelian group $G$ and I consider the group of the characters $G^*$, i.e the group of morphisms $\phi: G\to \mathbb{C}^*$ , then is it true that $\sum_{\chi \in G^*}\chi(h)=0$ ...
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1answer
41 views

If $G$ is a group of an even order, show that there exists an element of $G$ of order 2. [duplicate]

Im not sure where to start. I first assume $|G|<{\infty}$ and $|G|=2k$ with $k\ge{1}$. Then take some element $g\in{G}$, but I can't see what I can show from the assumptions.
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2answers
91 views

Order of $x$ with $x^p=a$

Using only elementary group theory. Let G be a group with an element $a$ of order $k$. Let $p$ a prime which divides $k$. If there is $x$ in G with $x^p=a$, show that $x$ has order $pk$. My ...
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0answers
24 views

Quotient by maximal normal locally finite subgroup

Any group contains a unique maximal normal locally finite subgroup (this is an exercise in Robinson, second edition, p. 436). Does the quotient by this subgroup have any notable properties? Edit: ...
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0answers
16 views

Show that graph $union$ and $join$ forms a semiring

Let $union$ $\cup$ and $join$ $\triangledown$ be two binary operations in a set $S$ of undirected simple graphs. If the self loops and the multiple edges formed as a result of these operations are ...
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0answers
41 views

Does there exist a finite group that is both perfect and immaculate?

A group $G$ is called perfect iff $G’ = G$. A finite group $G$ is called immaculate iff its order is equal to the sum of orders of its proper normal subgroups. Does there exist a finite group $G$, ...
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2answers
68 views

A question regarding the group $G = GL_2(\mathbb Z/ p\mathbb Z)$.

For any prime $p$ , consider the group $G =\mathrm{ GL}_2(\mathbb Z/ p\mathbb Z)$. Then which of the following are true? 1) $G$ has an element of order $p$. 2) $G$ has exactly one element of order $...
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1answer
28 views

Representation theory, group products can somebody please help me with this proof?

so the question is: (ii) Each irreducible representation of $G_1\times G_2$ is isomrophic to a representation $\rho^1\otimes \rho^2$, where $\rho^i$ is an irreducible representation of $G_i$ ($i=1,...
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0answers
28 views

Does $(\Bbb{R}, +)$ admit an irreducible $2$-traversal?

For a given natural number $k$, I'm going to call a subset $T$ of the plane $\Bbb{R}^2$ a $k$-traversal if, for any $x \in \Bbb{R}$, \begin{align*} k &= \operatorname{card} \{(a, b) \in T : a = x\}...
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1answer
159 views

Why is dim(ρW)=dimW for ρ being a linear representation?

first time asking on here so sorry if it's a bit clumsy. I will have to hold a seminar tomorrow and my professor helped me with one of my proofs. The proof is about showing that for any group $G$ and ...
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0answers
40 views

Group algebra and polynomial algebra.

It occurred to me that since the group algebra of $\mathbb{Z}$, $k\mathbb{Z}$, has multiplication $$\left(\sum _{n\in \mathbb Z} a_n n\right)\cdotp \left(\sum _{n\in \mathbb Z} b_n n\right)=\left(\...
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1answer
37 views

Normal subgroups of invertible affine transformations $\pmod p$

Fix a prime $p$. Find the number of normal subgroups of the group $G$ of invertible affine maps $x \to bx + c$, $b \neq 0$ on $\mathbb{Z}/p\mathbb{Z}$. It is clear that the group has cardinality $p(p-...