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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Are there groups of order $p^4q^2$ which are not semi-direct product?

It is easy to show that if $G$ is a group of order $p^2q^2$, where $p,q$ are primes with correspondings Sylow subgroups $P,Q$, that $G$ is a semi-direct product of $P$ and $Q$. Moreover, if $pq\neq ...
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How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
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G acts freely on X. G is paradoxical implies X is also paradoxical

I am proving the Banach-Tarski paradox using a series of small results. For definition of certain terms, see here. Group $G$ acts freely on $X$ i.e. $\operatorname{Stab}(x)=e, \ \forall \ x\in X$. ...
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Examples of irreducible representations

Which of the following representations are irreducible? 1) The tautological representation of $D_n$ on $\mathbb{R}^2$ 2) The action of $U(1)$ on $\mathbb{C}$ by multiplication 3) The ...
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2answers
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Group Actions: Verify a Bijective Correspondence

This is an old exam problem: Given an action of $G$ on $X$, we can define $\varphi: G \to S_X$ by the rule $\varphi(g) = \sigma_g$, where $\sigma_g$ is left multiplication by $g \in G$. Prove that ...
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Unit group of a field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isnt $k = ...
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24 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
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Aut$(G)\cong \Bbb{Z}_8$

I am looking for a group such that Aut$(G)\cong \Bbb{Z}_8$. Obviously Aut$(\Bbb{Z}_n)\ncong \Bbb{Z}_8$ for any $n$. Also Aut$(D_4)\cong D_4$, neither symmetric/alternating groups are of any help ...
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Existence of a special Central cyclic subgroup

I have two related questions. Let $G$ be a finite $p$ group. Can we always choose a central subgroup $N$ of order $p$ not contained in the commutator subgroup? Clearly we cannot do that for extra ...
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*$G$-invariant* symmetric bilinear form & $G'=\Bbb Z_2\times\Bbb Z_2$.

I got a problem with the last point I solved all the points, from (a) to (h), but I have no idea how to solve (i): how can I associate a bilinear form to a represtation? What is a $G$-invariant ...
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Condition of reversibility of Markov Chain [closed]

Show that a Markov Chain is time reversible iff $\pi _{i} P_{ij}= \pi _{j} P_{j i}$
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Tensor product of representations of a product group?

Given some group $G$ that can be written as product of two other groups $$G = G_1 \times G_2 $$ and some representation of this group written in terms of representations of $G_1$ and $G_2$ $$R = ...
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29 views

Is the product of group representations commutative?

Consider, for example, the product of $E_6$ representations $$ 78 \cdot \overline{351}_s \cdot 78 \cdot 351_s, $$ where the $s$ denotes symmetric. Is this equal to $$ 78 \cdot 78 \cdot ...
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Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
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Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
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Covering groups

I am studying the Steiner system $S(5,6,12)$ and the ternary extended Golay code $\mathscr{C}_{12}$. The automorphism group of the Steiner system is the Mathieu group on twelve elements $M_{12}$ ...
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1answer
28 views

Abelianization of a $p$ group.

Let $G$ be a $d$ generated finite $p$ group. Let $N$ be normal subgroup of order $p$ contained in $[G,G]\cap Z(G)$. Can we say say that $(G/N)/[G/N,G/N]=(G/N)_{ab}$ is a direct product of $d$ cyclic ...
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1answer
59 views

On cyclic decomposition of element in $S_n$

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...
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21 views

Proper formulation of one-to-one and onto proofs for group isomorphism

I have to construct an isomorphism for the two groups. I have the isomorphism itself but I'm not sure if my formulation is correct in regard to proving the mapping being 1-1 and onto and I don't want ...
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84 views

Example of non isomorphic groups with isomorphic group algebras

Below is the construction of two non isomorphic groups, $G_1$ and $G_2$ such that $KG_1 \cong KG_2$ for any field $K$. (My Doubts lie within.) Consider two groups $Q_1=\langle x_1,y_1,z_1\ |\ ...
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1answer
36 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
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1answer
115 views

A game from Exercise in Artin's Algebra (Chapter 2 M.13)

I found an interesting problem in Chapter 2 for Artin's Algebra (2nd Ed) in the Miscellaneous section that I haven't been able to figure out. The text of the problem is quoted below. M.13 (a ...
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Prove that $|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$ [closed]

Suppose that $G$ is a group of order $n > 1$, prove that $$|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$$ where $k=\lfloor \log_2 (n-1)\rfloor$.
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Derived subgroup of $S_n$ and $D_n$

I know that Derived/Commutator subgroup of $S_3$ is $A_3$ and commutator subgroup of $D_4$ is cyclic of order $2$. But What about derived groups of $S_n$ and $D_n$? How can I calculate them?
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2answers
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Necessary and Sufficient conditions to be a subgroup and/or a normal subgroup.

If $x \in G$, is it possible that $C = \{g^{-1}xg : g \in G \}$ is a subgroup of $G$? Can $C$ be a normal subgroup of $G$? (What are necessary and sufficient conditions to be such a subgroup?) ...
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The Stabilizer of the coset for the action of G on $G/H$ by left multiplication.

Let $H$ be a subgroup of $G$. What is the stabilizer of the coset $aH$ for the action of $G$ on $X=G/H$ by left multiplication? So, I think I've done this one correctly: The Stabilizer is of the ...
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0answers
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What's the asymptotic of the radius of the Rubik square Cayley graph?

This post is a sequel of The Rubik Square permutation groups, which should be read first to understand the notation. Question: what's the radius$^*$ of the Cayley graph of $G_n$ generated by the red ...
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Group Theory: Suggest video lecture (In English)

Please suggest video lecture for following topics in Group Theory. Revision of definition and examples of groups, subgroups. Cyclic Groups, Classification of subgroups of cyclic groups. Permutation ...
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The Rubik Square permutation groups

This post was inspired by this webpage of mathematical challenge due to Mickaël Launay (French). Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as for the following ...
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Group $G$ whose center $Z(G)$ is cyclic and with $G/Z(G)$ commutative

I have some issue to solve following exercise. The exercise is from a French book on Algebra (cours d'Algèbre) from Jean Querré. The book is from the 1970's. If the center $Z(G)$ of a group $G$ is ...
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1answer
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Number of Cosets of Intersection of Subgroups

Similar question has been asked on SE before but the problem statement is usually more specific and gives more information (in particular, tells you what to prove), but this problem asks to prove or ...
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1answer
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Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?

I saw this question on an old qualifying exam: Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$? A simpler problem would be to show ...
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A problem on order of a Group. [closed]

Let $G$ be a group of order $8$ and $x$ be an element of $G$ of order $4$. Show that $x^2 \in Z(G)$, the center of $G$. How this result can be proved?
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Proof that the kernel is a normal subgroup of the domain: repeated line

On proofwiki (https://proofwiki.org/wiki/Kernel_is_Normal_Subgroup_of_Domain), the lines corresponding to 'definition of identity' and 'definition of kernel' are identical. Why do we need the second ...
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Quotient of the upper-half plane as a projective line

In the discussion of the Belyi theorem here one pointed out that $\mathbb{H}^{*}/SL(2, \mathbb{Z})$ is isomorphic to the (complex) projective line $\mathbb{P}^1$, where $\mathbb{H}^{*}=\mathbb{H} \cup ...
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2answers
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Classifying groups of order $5 \cdot 11 \cdot 61$

My question is whether I'm classifying groups of order $5 \cdot 11 \cdot 61$ correctly. (This is a qualifying exam question, so I also want to make sure that I'm doing it “efficiently”.) Sylow's ...
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1answer
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The related concepts to a special statement

I saw the following statement later but I don't know it is true or not and I don't remember its reference: Suppose that $A$ and $B$ are non-empty sets and $G$ is a group with the generating set $A$ ...
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Finding the number of elements of particular order in the symmetric group

I know how to find the order of element in any group $G$, for example the order of $2$ in $\mathbb{Z}_5$ is $5$ as $2 + 2 + 2 + 2 + 2 = [10]_5 = 0 0$, which is the identity in $\mathbb{Z}_5$. But, ...
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2answers
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Prove image of symmetric group into additive group of real numbers is zero

Suppose $ f : S_{n} \rightarrow (\mathbb{R} , + , 0 , - )$ is a group homomorphism. Prove $ f(S_{n}) = {0} $, i.e., $f(\sigma) = 0$ for every $ \sigma \in S_{n}$with $n \geq 1 $ I cannot seem to find ...
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Prove that $|GL_n(\mathbb{F})|< q^{n^2}$.

Let $\Bbb F$ be a finite field, say $|\Bbb F|=q$; then we know that $|GL_n(\Bbb F)| < \infty$. But how can we prove that $|GL_n(\mathbb{F})|< q^{n^2}$? I'm guessing because there $n^2$ ...
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Find character table for symmetric group $S_3$

This group contains all permutations of 3 elements, so it has order 3!=6. Its three congruency classes are {1}, {(1,2),(1,3),(2,3)}, {(123),(132)}. As we know that the number of congruency classes ...
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Why is $U(10)\not\approx U(12)$?

I'm trying to understand this example on isomorphism but failing to do so. I know that if two groups have a differing number of elements of each order they are not isomorphic. I assume this is what ...
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$GL_n(\mathcal{F})$ is a finite group if and only if $\mathcal{F}$ is finite

Show that $GL_n(\mathcal{F})$ is a finite group if and only if $\mathcal{F}$ has a finite number of elements. These are my thoughts. The order of the group $GL_n(\mathcal{F})$ is ...
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1answer
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Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
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Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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Explain why D3 cannot be a subgroup of D8

To be a subgroup, a subset of a group must satisfy the group axiom but in this case, I do not see how the group axiom plays a part. Could someone explain to me why the above question is true?
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1answer
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What is the order of a rotation group

I was recapping some question on groups for my upcoming exams and chanced upon a question asking for the order of a rotation group $$C_{8}$$. Is the order of a rotation group 2n or $$\frac{2 ...
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2answers
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Quotients of Solvable Groups are Solvable

I just proved that subgroups of solvable groups are solvable. So given that $G$ is solvable there is $1=G_0 \unrhd G_1 \unrhd \cdots \unrhd G_s=G$ where $G_{i+1}/G_i$ is abelian and for $N$ a normal ...
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Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. [duplicate]

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. Prove that, $\exists$ $x\neq e$ such that $x^2=e$ where $e$ represents the identity of $G$.
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Groups and morphisms making a lattice structure

(1) Let X and Y be groups. Write mono:X->Y and epi:X->Y (resp.) to denote the existence of a monomorphism or epimorphism from X to Y (resp). mono:X->Y would mean that there is a subgroup of Y ...