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 studies groups.

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Problem from Armstrong's book, “Groups and Symmetry”

I haven't gotten all that far with this: If $a$, $b$ are members of the permutation group $S_n$, and $ab=ba$, prove that $b$ permutes those integers which are left fixed by $a$. Show that $b$ must ...
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2answers
29 views

If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)

The following is from Hungerford's graduate algebra book, Chapter 1, Section 4, Proposition 8. The proposition states that: If $H$ and $K$ are subgroups of a group $G$, then $[H:H \cap K] \leq [G:K]...
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Is the group of units in $\mathcal{O}_{\mathbb{Q}(\sqrt{2})}$ finite and cyclic?

Let $K=\mathbb{Q}(\sqrt{2})$ and let $\mathcal{O}_K$ be its ring of integers. Consider the group of units in $\mathcal{O}_K$. Is it finite? Is it cyclic? My thought: For any $\alpha = a+b\sqrt{2}$, ...
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Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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Conjugacy-classes of cyclic subgroups [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p^m\mathbb{Z})$ of cyclic subgroups of order p?.
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37 views

Group Theory of Friends [on hold]

How group theory applies on five friends. friends = {f1, f2, f3, f4, f5} f1 and f5 form subgroup. f2 and f3 are anti. who is the identity? and how association works. What example can be set to ...
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1answer
557 views

groups with same number of elements of each order

When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...
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57 views

A question about automorphism groups of finite groups

I’ve encountered the following question whilst helping a colleague study for comprehensive exams, and I’m stuck on it: Let $G$ be a finite group such that the natural action of the automorphism ...
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2answers
41 views

How to determine if certain operation is associative based on Cayley table

I have the following table and I don't know how to determine if an operation is associative based on the table. Is there an easy way to do it? Or it's just brute force \begin{array}{|c|c|c|c|c|c|} \...
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25 views

Pronormal subgroups of direct products

Suppose that $G = A \times B$. Let $U = A \times \pi_B(U) \leq G$ such that $\pi_B(U)$ is pronormal in $B$. Then $U$ is pronormal in $G$. This is part of a proof of Proposition 4.3 in Pronormal ...
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28 views

What is conjugacy class of group of motion of the plane?

This is a question from Artin. I'm unable to even begin to solve it. I don't understand what does it mean? Do we have to find conjugacy class of $D_n$ ? If so, then how?
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37 views

Decomposition of a group into disjoint subgroups

This is an old exam question and I have no idea how to answer it: Can a group be decomposed into a disjoint union of subgroups?
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1answer
34 views

About the notation of composition of permutations in Lang's book

In Lang's "Algebra", p.30-31, I'm confused about the order of reading the composition of two permutations. In p.30, it seems that we read it from left to right (see the bottom equations), but for p.31,...
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1answer
22 views

Is there a standard notion of a group “biaction”?

For a group $G$ there is a natural notion of left action on subsets of $G$ given by $g \triangleright H := gH$. But simultaneously, there is a natural right action as well: $H \triangleleft g := Hg$. ...
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1answer
63 views
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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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53 views

Show that $H$ is not a subgroup of the group $G=\mathbb{Z}_{24}$ (with operation +)

Let $G=\{0,1,2,...,23\}$, and I know that a subgroup is closed underneath the operation and has the identity element. So in order to show it isn't a subgroup, do I just have to show that $$H=\{a + 24\...
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1answer
138 views

Is there a rank 3 boolean interval [H;G] with G simple group?

Let $[H;G]$ be an interval of finite groups. The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. For example, for $m = p_1p_2 \dots p_n$ square free, $[1;\mathbb{...
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2answers
53 views

If $gh = hg, \ \ \gcd(|g|, |h|) = 1$, then $|gh| = |g||h|$($|a|$ is the order of element $a$ in a group $G$)

Let $G$ be a group and $g,h \in G$. I need to prove that if $g$ and $h$ commute and their orders are coprime, then $|gh| = |g||h|$, that is, the order of their product is the multiple of their orders. ...
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1answer
15 views

If $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$

Question In my group theory course, I am asked to show for $\sigma \in S_n$ that if $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$. My answer Let $\sigma \...
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1answer
59 views

How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
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1answer
54 views

Every group of order $567$ has normal subgroup of order $27$.

Prove that every group of order $567$ has a normal subgroup of order $27$. Let $G$ be such a group. Then $|G| = 3^4\cdot7.$ Let $H\in\text{Syl}_3(G).$ From the Sylow theorems, we have that $n_3 | 7, ...
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1answer
20 views

Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
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Number of $p$-Sylow subgroups in $D_{2n}$

Let $2n = 2^ak$, where $k$ is odd. We wish to show that the number of $2$-Sylow subgroups of $D_{2n}$ is $k$. My approach has been to construct such a $2$-Sylow subgroup $P_2$, and then show that the ...
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Conjugacy-classes in $\operatorname{GL}_n(\mathbb{Z}/p^m\mathbb{Z})$ [duplicate]

Enumerate the number of conjugacy classes in $\operatorname{GL}_n(\mathbb{Z}/p^m\mathbb{Z})$ of cyclic subgroups of order $p$?.
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3answers
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Every permutation is a product of two permutations of order 2

I am trying to solve a problem, not for homework, and it has me stomped! For $n\geq 4$ and $\alpha\in S_n$, $$\alpha=\dot{\alpha}\dot{\beta}$$ where $\dot{\alpha},\dot{\beta}$ are of order 2. I know ...
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1answer
15 views

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite?

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite? Consider for instance a finite sequence of moves (...
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$R$ be a commutative unital ring , is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+)$ and $(R^{\times} ,\cdot)$ are not isomorphic as groups ? I know that the ...
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Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
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$|G|=p^kq$, $p,q$ prime, $p^k\leq 2q$ implies one of the Sylow subgroups is normal

I could use some help with this one: Let $G$ be a group, $|G|=p^kq$ where $p,q$ are prime and $p^k\leq 2q$. Prove that at least one of the Sylow subgroups is normal.
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Group of order $105$

It is clear that a finte group of order $105$ is not simple since it contains a normal Sylow $7-$subgroup or a normal Sylow $5-$subgroup. I wonder if it is possible that there exists a Sylow $7-$...
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A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
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Group von Neumann algebras

I have a question about group von Neumann algebras structure. If $L(G)$ is a subset of $L(H)$, can we find a subgroup $G_1$ of $H$ such that $L(G_1)$ be isomorphic to $L(G)$? I appreciate any help.
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Group of order 175 is Abelian

Question: Prove that any group of order 175 is Abelian. The solution: I am unable to understand why the intersection of the normal subgroups is the trivial intersection. Any help is ...
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1answer
47 views

Why is there no retraction between $D^2 \times S^1$ and $S^1\times S^1$?

Why is there no retraction between $D^1 \times S^1$ and $S^1\times S^1$? I have no idea how to prove. I just know that if there is then $\mathbb{Z}\times \mathbb{Z}$ contains a copy of $\mathbb{Z}$. ...
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Different Representation Matrices from same Generating Set

Motivation: This post. $K \subset S_n, \langle K \rangle =G \leq S_n$. We can create a Representation Matrix $M$ from $K$ that represnts $G$ (Furst. Hopcroft, Luks). Question: Is $M$ unique for a ...
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$\{g\in G\mid\alpha(g)=g^{-1}\}=\frac34|G|$, find an abelian subgroup of index 2

$G$ is a finite group, $\alpha$ is an automorphism of $G$ and $I=\{g\in G\mid\alpha(g)=g^{-1}\}$. If $|I|=\frac34|G|$, show that $G$ has an abelian subgroup of index 2. Related question I don't ...
2
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1answer
27 views

Homomorphism between S4 and A4

I'm asked to find a group G with a subgroup H such that there is no normal subgroup N of G which performs: G/N =~ H. I thought of G=S4 and H=A4, because I don't think there is an homomorphism from S4 ...
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1answer
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find torsion coefficients of groups

I have to find torsion coefficients of groups $G_1\simeq Z/2\oplus Z/4\oplus Z/3\oplus Z/3\oplus Z/9$ and $G_2\simeq Z/15\oplus Z/20\oplus Z/18$. I want to ask if my calculations are correct. For $...
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If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
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1answer
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possible cyclic group from fundamental theorem of finite abelian

Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic? By the Fundamental theorem of finite abelian group: $\left | G \right |=225=3^{2}...
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Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
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Isomorphism between semidirect products

Alperin and Bell, Groups and Representations, section 2, Proposition 11, p. 23, states: "Let $H$ be a cyclic group and let $N$ be an arbitrary group. If $\varphi$ and $\psi$ are monomorphisms from $H$ ...
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3answers
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Does the converse of Lagrange's theorem hold for any finite Dedekind group?

I'm currently reading a book on algebra and the following argument is used to prove that the converse of Lagrange's theorem (if $d$ divides $|G|$ there exists a subgroup of order $d$) holds for any ...
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Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
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2answers
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Normal closure of a subgroup of a free group.

Let $G$ be a finitely generated free (nonabelian) group, $H$ a subgroup generated by some of the generators of $G$, and $a: G\to AG$ be the projection to the abelianization $AG:=G/[G,G]$. Is it true ...
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Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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How can I prove that the the number of elements of order $k$ in $\mathbb{Z}_n$ is φ(k)? [on hold]

How can I prove that the number of elements of order $k$ in $\mathbb{Z}_n$ is ϕ(k) where $k$ is number that divides n ?
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If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
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Error Correcting Code and Graph Theory

I am currently in an introductory graph theory class, and we are supposed to give a short presentation by the end of the semester. Recently, I've learned (a very small amount) about error correcting ...
18
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Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, ...