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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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Regular normal subgroups.

Hi: I don't understand the proof ($G$ is a finite group). From the equality I get $\beta=\alpha^{{(x_{\beta})^g g^{-1}}}$. Unfortunately the exponent does not necessarily belong to $N$ and so I can't ...
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Meaning of notation $St_{\mathcal{B}}(b)/H$ in Peter May's Concise Course in Algebraic Topology

I'm reading Peter May's Concise Course in Algebraic Topology, and currently he is describing the construction of coverings of groupoids. I'm having some trouble understanding the notation $$St_{\...
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Confusion about generators in $SU(2)$ (Lie algebras)

I have just started learning about Lie algebras and got confused with this: Some sources say the generators are $J_0,J_1$ and $J_2$ and some use $J_0,J_+$ and $J_-$. Which set is correct? Or if both ...
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Show that if $G$ is a finite group and ${\rm Aut}(G)$ is cyclic, then $G$ is abelian. [duplicate]

This question seems a little too straightforward, so I'm just suspicious that I'm overlooking something. I know that if $G/Z(G)$ is cyclic, then $G$ is abelian ($Z(G)$ being used to denote the center ...
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Examples of Subgroups [closed]

Suppose that x is an element of a group $𝐺$ with $|x| = 5$. Prove that $𝐢(x^3) βŠ† 𝐢(x)$ , where $𝐢(x)$ denotes the centralizer of the element $x ∈ 𝐺$. How about if $|x| = 4$ ? Use Definition ...
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elmentary properties of groups [closed]

Consider the group 𝐷15. (a) Find | 𝐷15|. (b) How many elements π‘₯ in 𝐷15 satisfy the equation π‘₯^2 = 𝑒, where e is the identity element in 𝐷15? Explain. (c) What are the elements in 𝐷15 with ...
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1answer
31 views

Show that $gK = Kg$ for every $g \in G$ knowing that is true for every coset representative.

The original question is this: Let $G = S_4$ and let $K = \{ 1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$. Determine the cosets of $K$ in $G$. Conclude that $gK = Kg$ for all $g \in G$. I used sagemath ...
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Operation on the group is well defined “in the first and second factor”.

In the book "Algebra: Chapter 0" (Paolo Aluffi), Chapter II.7 (Quotient groups), it says that given an equivalence relation $\sim$ on a group $G$, for the operation to be well-defined "...
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24 views

Indecomposable integral representations of a group of order 2

This question is a duplicate of that 2010 MathOverflow question. I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$. ...
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Centralizer of a in G [closed]

Suppose that 𝑦 is an element of a group 𝐺 with |𝑦| = 5. Prove that 𝐢(𝑦^3) βŠ† 𝐢(𝑦), where 𝐢(𝑦) denotes the centralizer of the element 𝑦 ∈ 𝐺. How about if |𝑦| = 4?
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(123) is not cube of any element in symmetric group $S_{n}$ [duplicate]

This particular question was asked in masters exam for which I am preparing . Let n$\geq$ 3 be a natural number . Prove that the 3 cycle (1,2,3) is not a cube of any element in symmetric group $S_{n}$...
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1answer
37 views

Compute the Group of Automorphisms of the Abelian Group $\mathbb{Q}^n$

Compute explicitly the group of automorphisms $\textrm{Aut}(\mathbb{Q}^n)$ of the abelian group $\mathbb{Q}^n=\mathbb{Q}\times ...\times \mathbb{Q}$. This task is from on an old exam. As far as I ...
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Classify all finite groups $G$ up to isomorphism with the following property; For any two subgroups $K,H \leq G$, either $H\leq K$ or $K\leq H$

At a bit of a loss on how to even get started here. I know that any $G$ with this property will have to be cyclic, but it's definitely not all cyclic groups. Any thoughts would be greatly appreciated. ...
2
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1answer
42 views

$\delta$ thin trianges implies solvable conjugacy problem for hyperbolic groups: Confusion about $\delta$-rectangles

I am trying to understand the proof that a linear dehn function implies solvable conjugacy. I am referring to Notes on solvable and automatic groups by Michael Batty, after Panagiotis Papasoglu. Here ...
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1answer
32 views

Find smallest positive integer such that this particular group is isomorphic to subgroup of $S_{n} $

This is a problem of a masters exam for which I am preparing. Find the smallest positive integer n such that $\mathbb{Z}/2\mathbb{Z} $ Γ—$\mathbb{Z}/2\mathbb{Z} $Γ—$\mathbb{Z}/2\mathbb{Z} $ is ...
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1answer
67 views

Group of order 27

Let G be a group of order 27. Let $x\in G/Z(G)$ where $Z(G)$ is the center of $G$. Then what is the possible orders of the centralizer of $x$ and what is the number of conjugates of $x$? Any hint or ...
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Why is this not a group?

It's been a while since I took abstract algebra but I'm wondering what is wrong with my reasoning here A group $(G, \circ)$ is defined as a tuple that consists of a set $G$ along with an operation $\...
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39 views

Normality of subgroups isn’t transitive

I’m trying to disprove the following statement: β€œlet $G$ be a group, and suppose $H,K$ are subgroups of $G$; if $H$ is normal in $G$ and $K$ is normal in $H$, then $K$ is normal in $G$.” My ...
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Normalizer of normalizer of Sylow p-subgroup

The question was to show that $P\in Sly_p(G)$ then $N_G(N_G(P))=N_G(P)$, where $G$ is a group. I did it using the Sylow C-theorem but was wondering if it was possible to do the problem without ...
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Clarification on a lemma for Fundamental Theorem of Finite Abelian Group

The lemma: Let $G$ be a finite abelian $p$-group and suppose that $g\in G$ has maximal order. Then G is isomorphic to $\langle g\rangle\times H$ for some subgroup $H$ of $G$. The proof I am reading is ...
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Meaning of $H$ normalizing $N$. [duplicate]

Let $G$ be a group and $H$ and $N$ be subgroups of $G$. I wonder what the canonical meaning of "$H$ normalizes $N$" in literature is: (1) $\forall h\in H$, $h N h^{-1}=N$. (2) $H=\{h\in G: ...
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Let $N$ be an abelian minimal normal subgroup of $G$. Then $N$ has a complement in $G$ iff $N\not\leq \Phi(G)$.

Let $N$ be an abelian minimal normal subgroup of a finite group $G$. Then $N$ has a complement in $G$ iff $N\not\leq \Phi(G)$, where $\Phi(G)$ is the Frattini subgrup of $G$. The Schur-Zasenhaus ...
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3answers
121 views

Can a group with elements $I,C,L,X$ have $CL$ as an entry in its Cayley table?

I am attempting to understand a certain phenomenon (which I do not currently understand well enough that explaining it would add value to the question) for which I have reasoned out the following ...
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34 views

Cardinality of general polynomial equations over finite fields

Let $E: Q(y)=P(x)$ be an equation over a finite field $\mathbb F_p$ given any prime $p$, and any polynomials $Q(y)$ and $P(x)$ (polynomials in $y$ and $x$ respectively). Is there a general approach to ...
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51 views

The sum over the elements of a multiplicative subgroup of a field is always zero?

Let $G \leqslant K^{\times}$ be a multiplicative subgroup of the group of units of a field $K$. Then $\sum_{g \in G} g = 0$ necessarily. But, how can we prove that $\sum_{g \in G} g$ is always an ...
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129 views

What groups have a homomorphic image $\Bbb{Z}/2\Bbb{Z}$?

Is there a simple characterization for all the groups $G$ so that there exists an epimorphism $\varphi:G\to\Bbb{Z}/2\Bbb{Z}$? First assume there exists a nontrivial homomorphism $\varphi:G\to\Bbb{Z}/...
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Order of the group of commuting elements in a finite group

Let $G$ be a finite group of order $n$ and class number $k.$ Show that $$ \left|\left\{(a,b)\in G^2: ab=ba\right\}\right|=nk. $$ I considered $G$ acting on itself by conjugation, and then applied ...
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If $H$ is a subgroup of a finite abelian group $G$, then $G$ has a subgroup that is isomorphic to $G/H$.

I know Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$? has two answers. I don't understand how the first answer works and I have doubt about that answer. The second ...
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1answer
34 views

Does the coinvariant algebra ever admit nilpotent elements?

Here's a simple question, presumably the answer is well known to certain people. Let $k$ be a field and let $A$ be a finitely generated $k$-algebra which is reduced (i.e. it admits no nilpotent ...
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1answer
58 views

These factor groups are isomorphic to which group

I am asking a question from Abstract Algebra Assignment in which I am having a trouble . Let $$G=\left\{\begin{pmatrix}a&b\\0&a^{-1} \end{pmatrix}: a,b\in\mathbb{R} , a>0\right\}$$ and $$N=...
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1answer
52 views

Proving that a group is abelian!

Supposing we have a group $(G,*)$ (Which we don't know the elements of). It is given that for each $x, y \in G$, exists $x*y*x=y$. That's the only information we receive, How can we prove that group G ...
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How the square model of Fano 3-space, a.k.a. $PG(3, 2)$, embeds in the MOG?

I'm studying deeply the Curtis' MOG original article, i.e. a New combinatorial approach to $M_{24}$ and I'm still struggling a bit in reading the final 35 6x4 matrices representing the MOG. Actually ...
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1answer
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Defining a quotient field of a field using multiplicative subgroup possibly?

Let $H \leqslant K^{\times}$ be a multiplicative subgroup of the group of units of a field $K$. It is normal since we're in a field (a commutative ring in particular). The multiplicative cosets $K/H = ...
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Lemma for Fundamental Theorem of Finite Abelian Groups

I am trying to understand the following lemma for FTOFAG In the book, The proof is by induction： We proceed by induction on $|G|$. Clearly, the case where $|G| = p$ is true. Now suppose the statement ...
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Equations over finite fields to prove primality

Inspired by the Eliptic Curve Primality Test, and classical primality tests, I wanted to know if any particular equation (using multivariate polynomials) over finite fields. The group $(\mathbb Z/n\...
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An abelian group $G$ that has order 216, but $6G$ has order 6

I’m trying to prove the following statement: β€œif G is a finite abelian group with $|G|=216$ and $|6G|=6$, then determine $G$ up to isomorphism.” I know that by the fundamental theorem of finite ...
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1answer
38 views

Let $H\le G$ of index $3$. Prove that either $H\unlhd G$, or that $H$ has a subgroup $N$ of index $2$ in $H$ such that $N\unlhd G$.

Let $H$ be a subgroup of a group $G$ of index $3$. Prove that either $H$ is normal, or that $H$ has a subgroup $N$ of index $2$ in $H$ such that $N$ is normal in $G$. All I could show is that if $H$ ...
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1answer
22 views

Understanding a result regarding subgroups containing the $n$-th powers of a field as a subgroup of finite index

Let $F$ be a field which contains a primitive $n$-th root of unity. For any Galois extension $E$ of $F$, define $B(E) = F^\times \cap E^{\times n}$. Now let $H$ be a subgroup of $F^\times$ containing $...
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How to find the splitting of a polynominal modulo $p$ with Dedekind's Theorem? [closed]

I don't understand how I can calculate the factors of $f$ modulo the primes. For example: Let $f(X)=X^6+X^4+X+3$. Here are the factorizations of $f(X)$ modulo the first few primes: $$ f (X) \equiv (X+...
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Understanding the map $F^\times \cap E^{\times n} \to H^1(G,\mu_n)$

Let $F$ be a field and $\zeta \in F$ be a primitive $n$-th root of unity. Also, let $E/F$ be a finite Galois extension with group $G$. Now I would like to understand the map $f: F^\times \cap E^{\...
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1answer
30 views

Cyclic groups as normal sobgroups of SO(3)

I have to argue why none of the cyclic groups $C_n$ is a normal subgroup of $SO(3)$. Nevertheless, I haven't found an argument for that. First I used the definition of normal subgroup: If some $C_n$ ...
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23 views

Kummer Theory: Understanding the isomorphism $F^\times \cap E^{\times n}/F^{\times n} \to \operatorname{Hom}(G,\mu_n)$

Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$. Now I am trying to understand the following section from Milne's Fields and Galois Theory (page 73): The part "...
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Understanding the map about the classification of all abelian extensions with Galois groups with a fixed exponent (Kummer Theory)

Let $F$ be a field and let $\zeta$ be a primitive $n$-th root of unity in $F$. Also, let $E/F$ be a finite Galois extension with Galois group $G$. Now I am trying to understand the following Theorem ...
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2answers
52 views

Order of $(1 \,3)(2 \, 5 \, 4)$ in $S_5$

What is the order of $(1 \,3)(2 \, 5 \, 4)$ in $S_5$? From number theory, I remember that we defined the order to be the smallest positive integer $k$ for which $a^k \equiv 1 \pmod{n}$ and also $a$ ...
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1answer
19 views

Interleaving a fixed element into a given sequence of elements of a permutation group and the image of a point

Let $G \le \operatorname{Sym}(n)$ be a permutation group on $\{1,\ldots,n\}$. For $\alpha \in \{1,\ldots, n\}$ and $x \in G$, write $\alpha^x$ for the application of $x$ to $\alpha$. Fix some $g \in G$...
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1answer
57 views

The proof of $S_n \cong A_n \rtimes \{e, (12) \}$

$\blacksquare~$ Problem: Let $G = S_n, H = A_n$ and $K = \{ e, (12) \}$. Show that $S_n \cong A_n \rtimes K$. $\blacksquare~$ My Approach: Let $G = S_{n}$. Where $S_{n}$ is the symmetic group of ...
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57 views

A family of groups as a monoidal category

1.Context My lecture notes present the following example of a monoidal category: Let $G:=(G_n)_{n\in \mathbb {N_0}}$ be a family of groups with $G_0$ the trivial group with one element. We define a ...
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35 views

How to prove that pqr order group is solvable group？ [duplicate]

Let $G$ be a group and $|G|=pqr$, where $p,q,r$ are prime numbers that are not necessarily distinct. Show that $G$ is solvable. I try to discuss the classification of groups of order $pqr$ and I also ...
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1answer
33 views

Multiplying cycles

How does one perform cycle multiplication? It seems that every textbook I read has a different notation for this and it's not clear at all. Suppose I have $(123)(134)$ now some book stated that the ...
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An ascending union of reduced FATR groups is FATR

My question is taken out of a proof in the book "Infinite Soluble Groups" by Robinson and Lennox. Let me paste the proof first: Some reminders: a soluble group $G$ has FATR (finite abelian ...

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