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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Walk generating function for adjacency matrix

In its book "Random Walks on Infinite Graphs and Groups", W. Woess defines the Green function (or walk generating function) as $$G(x,y|z) = \sum_{n = 0}^{\infty} p^{(n)}(x,y)z^n, \quad x, y \in X, z \...
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2answers
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Injective map of abelian group and product of cyclic quotients

Let $A$ be an abelian group. We have a map from $A \to \prod{(A/I)}$ where $A/I$ varies over the cyclic quotients of $A$. This map is given by sending $x$ to $\prod (x \text{ mod } I)$. Is this map ...
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33 views

Cocycles and group extensions

I'm trying to understand how elements in the second cohomology group with coefficients in some other group correspond to group extensions. This is what I understand: Suppose we have two (countable) ...
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1answer
48 views

Why does $C_G(g)$ have index $q$ if $|G| = pq$ and $|g| = p$ for all nonidentity $g \in G$?

I'm studying Abstract Algebra, 3rd Edition by Dummit and Foote. In Section 4.4 (Automorphism), one of the examples proves that $G$ is abelian if both of the following hold. $|G| = pq$ where $p$ and $...
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2answers
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A finite union of infinite cyclic subgroups of a group $G$ is never a group.

$\mathbf{Question}$: Consider an infinite group $G$. Let $\langle a_1\rangle , \langle a_2\rangle, \ldots, \langle a_m\rangle $ ($\langle a_i\rangle =C_i$) be a finite collection of infinite cyclic ...
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1answer
23 views

If $r = \frac{1}{|G|} \sum_{g \in G} \chi(g^{-1})g$, then $r^2 = 1/(\chi(1)) r$ in the group algebra

Suppose we have a finite group $G$ and an irreducible character $\chi$ of $G$. Now, define in $\mathbb{C} G$ (the group algebra / group ring of $G$), the element $$r = \frac{1}{|G|} \sum_{g \in G} \...
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Help with disjoint cycle decomposition and finding image/pre-image of a S8 group

i have this exercise and i'm confused about the part where i find the image and pre-image. I solved it for 5 in alfpha, found the image and preimage (preimage is number that 5 results from and image ...
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1answer
48 views

Finding a Group isomorphism G

According to wiki, Isomorphism: Given two groups $(G, ∗)$ and $(H, {\displaystyle \odot })$, a group isomorphism from $(G, ∗)$ to $(H, {\displaystyle \odot })$ is a bijective group homomorphism ...
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1answer
22 views

Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram

I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$: First definition Given $\sigma \in S_n$ ...
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1answer
53 views

$\varphi: M \to N$ is onto if and only if for all $R$-mod homomorphisms $\alpha : F \to N$ there exists an $R$-mod homomorphism $\beta : F \to M$

Let $R$ be a ring, $F$ a nonzero free $R$-module, and let $\varphi : M \to N$ be a homomorphism of $R$-modules. Prove that $\varphi$ is onto if and only if for all $R$-module homomorphisms $\alpha : F ...
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What is the fibered coproduct of abelian groups?

From wikipedia: I am not sure I understand this. For example: Suppose we have $\alpha: \mathbb Z/(3) \to \mathbb Z/(6)$ where $$0\mapsto 0, \quad 1 \mapsto 2, \quad 2 \mapsto 4,$$ and we have $\beta:...
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Beautiful Frieze Groups [closed]

Maybe this is more of an art question than a math question but here it goes: What are the most beautiful themed (all having similar style) examples of all the 7 Frieze groups that you've seen? Here'...
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21 views

Unique cyclic subgroup in dihedral group is characteristic [duplicate]

I have seen a proof of "the cyclic subgroup H of dihedral group $D_{n}$ where n>=3 is characteristic". In that proof a result was used. " H is the unique cyclic subgroup generated by an element of ...
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31 views

Reduced multiplicative residue modulo p [duplicate]

I would like if someone could provide or show me where I can find the proof that $\mathbb{Z} _n^*$ is cyclic when n is prime. In particular I'm after a simple proof that involves fermats little ...
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To change deficiency of a group presentation

I am reading the paper "Virtual knot groups by Se-Goo Kim". In the proof of Lemma 2, the author first considered a group presentation $\langle t_1, \ldots, t_n~|~r_1, \ldots, r_m \rangle $, where $m=...
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In what type of morphism do sequences over a subset of a Prufer Group, have the same group structure as the group?

Let $G$ be the Prufer 2-group. Let a set of infinitely many sequences $S$ of infinite length range over the elements of $G$. Now suppose the subset of these sequences ranging over $G\setminus3G$ is ...
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61 views

Some definition in finite group theory

I am reading an article about number of sylow subgroups. I have this theorem and don't understand some definitions which I am in italics below . $\textbf{Theorem.}$ Suppose that $A$ is a fi nite ...
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2answers
54 views

Commutator power lying in commutator subgroup

The argument used in the proof of Proposition 3 of this math.SE answer appears to prove the following claim: Let $G$ be a group and let $H\subseteq G$ be a normal subgroup. Let $n\geq 0$ and let $x,...
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0answers
34 views

Coordinate Changes in Group Presentations [on hold]

Let $G$ be a finitely presented group with presentation $\langle x_0, \dots, x_n \; | \; R(x_0) \rangle$. Let $f$ be an automorphism of $G$ such that $f(x_0), x_1, \dots x_n$ generates $G$. Is it true ...
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+350

Dimension of pro-$p$ group

Problem. Let $G$ be a pro-$p$ group of finite rank. Prove that $$\mathrm{dim}(G) = \lim_{k \to \infty}\frac{\log_{p}|G:G^{p^k}|}{k}.$$ By definition $$\mathrm{dim}(G) = \mathrm{d}(H)$$ where $H$ is ...
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Why does cocycle condition imply a group?

In group cohomology, the 2-cocycle condition emerges from associativity (see e.g. here). From the answer to a previous question we see (at least for normalized cocycles) that the cocycle condition ...
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1answer
66 views

Find group of automorphisms

In group $ GL(2,R) $ I have subgroup H, generated by 2 elements: $ a=\begin{pmatrix} 1 & 2 \\ 0 & 1\\ \end{pmatrix} $ , $ b=\begin{pmatrix} 1 & 3 \\ 0 & 1\\ \end{pmatrix} $ I ...
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1answer
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Proving $G\simeq G^{op}$ as categories

Exercise 1.2.23 https://arxiv.org/pdf/1612.09375.pdf 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 ...
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2answers
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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
43 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
39 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
57 views

Subgroup of order $10$ must be normal? [closed]

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
45 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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0answers
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If $\phi(fg)=\phi(f)g$, how to call it? [closed]

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
47 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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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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63 views

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

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
53 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
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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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1answer
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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
54 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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+150

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 subgroup. 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
97 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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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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64 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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26 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
55 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
152 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
32 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
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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
98 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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13 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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0answers
25 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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0answers
46 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
31 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 ...