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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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$[G:H]<\infty$. Show that there exists a normal subgroup $N$ of $G$ such that $N \subset H$, $[G:N]<\infty$

I am reading "An Introduction to Algebraic Systems" by Kazuo Matsuzaka.(By the way, Kazuo Matsuzaka was one of the most famous elementary mathematics writers in Japan.) There is the following ...
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22 views

Find all homomorphism from $S_4$ to $z_m$.

Find all homomorphism from $S_4$ to $z_m$. The only normal subgroup of $S_4$ are trivial , entire group, $A_4$ and klein 4 group. If m is odd , then there is only one homomorphism (trivial)... What ...
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1answer
30 views

Regular permutations

$ \sigma \in S_n $ is a regular permutation if there exist disjoint cycles $ \tau_1...\tau_r $ of length m such that $ \sigma = \tau_1 \circ ... \circ \tau_r $ and $supp(\tau_1)\cup...\cup supp(\...
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1answer
70 views

When does there exist a section of $GL_n(\mathbb Z_p) \rightarrow GL_n(\mathbb F_p)$?

There is a reduction map $f: GL_n(\mathbb Z_p) \rightarrow GL_n(\mathbb F_p)$ for any prime $p$, when does there exist a group homomorphism $g: GL_n(\mathbb F_p) \rightarrow GL_n(\mathbb Z_p)$ such ...
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31 views

Group theory problem on counting the elements [on hold]

Let $G$ be a finite group and let $n$ be a factor of $|G|$.Then the number of elements of $G$ which satisfy the equation $x^n=1$ is a multiple of $n$. Help me understand this problem in elementary way....
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17 views

$D_4 \times \mathbb{Z}_2$ different upper and lower central series

I wanted to find a group $G$ which had different upper and lower central series. Moreover, both different to one of his central series. I have found that $D_4\times \mathbb{Z}_2$ is one of the ...
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13 views

When does $C(a^i)=C(a^j)$ if $|a|$ is finite?

I am looking for a necessary and sufficient condition for $C(a^i)=C(a^j)$ to hold in a general group G given that $|a|$ is finite and known. What if $|a|$ is a prime? Here $C(a)=\{g\in G \;| \;gag^{-...
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0answers
16 views

Fourier transform of the Klein group

Obviously we have the elements of $G = \lbrace e, a, b, c\rbrace\;$ where $\;c=ab$ define: $$\chi : G \rightarrow Z=\lbrace z\in C : |z|=1\rbrace $$ How would I construct the transform for each of ...
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2answers
52 views

Why study “virtual properties”?

In group theory, I have seen some results which involve "virtual properties". E. g. virtually abelian, virtually solvable etc. The definition is, according to Wikipedia (https://en.wikipedia.org/wiki/...
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1answer
20 views

proving that a group G has in order equal to a power of 2

I was working on a finite group $G$ in which every element different from the identity element has an order of $2$, and my goal is to prove that $|G|=2^r,r\in\mathbb{N}$. I did find a proof and I want ...
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1answer
32 views

Show that $\bigcap_{a \in G} a H a^{-1}$ is a normal subgroup of $G$.

I am reading "An introduction to algebraic systems" by Kazuo Matsuzaka. In this book, there is the following problem. I think this problem is easy but a little abstract for me. Is my answer ...
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0answers
29 views

If $\lim_{n \rightarrow +\infty} \frac{|T'_n|}{|T_n|} = 1$, then $\lim_{n \rightarrow +\infty} \frac{|T_n \Delta T'_n|}{|T_n|} = 0$

Let $\{T_n\}_{n \in \mathbb{N}}$ be a Folner sequence and $\{T'_n\}_{n \in \mathbb{N}}$ be any sequences of subsets of $\mathbb{Z}^d$ such that $\lim_{n \rightarrow +\infty} \frac{|T'_n|}{|T_n|} = 1$. ...
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1answer
33 views

If $\operatorname{class}(G) = 2$ and $\exp(G) = 4$ then $\exp(G') = 2$?

Let $G$ be a finite $p$-group. I'd like to prove (or disprove) that if the nilpotency class of $G$ equals two (i.e., $1 \neq G' \le Z$, where $Z$ is the center of $G$) and the exponent of $G$ equals ...
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43 views

Assume $a, b \in$ a group $G$. Given the identity $ab = ba^{-1}$ Find… [on hold]

Find $m, n \in \mathbb{Z}$ such that $ba = a^mb^n$. I've been attempting this question for about an hour and have only gotten that $ba = abaa$. What am I missing?
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3answers
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Does the order of $2$ in $Z_p$ divide the order of $2$ in $Z_n$?

Givens. Consider $(Z_n, \times)$ the group of integers coprime with $n = pq$, where $p, q$ are prime numbers. Similarly, $(Z_p, \times)$ is the group of integers $\{0, 1, 2, ..., p - 1\}$ coprime ...
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1answer
54 views

How is a group visually represented?

I don't understand an example on the wkipedia article for Lie groups: The group given by $H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi ...
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2answers
65 views

multiplying two normal subgroups is still normal?

Let $G_i \triangleleft G_{i+1}$ both subgroups of $G$. Let $N$ be a normal group. Does $G_iN \triangleleft G_{i+1}N$? Does $(G_iN/N) \triangleleft (G_{i+1}N/N)$? I know that $q:G\longrightarrow G/...
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1answer
74 views

Proving that $\text{Sym}(\emptyset)$ is a group without treating functions as sets

Consider the statement: "For each set $A$, there exists a function from the empty set to $A$." Can we determine whether this statement is true or false, without treating functions as sets (i.e. ...
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Discrete Spherical Symmetry Group

Take two spheres each having a certain number (say 5) of identical dots on them. What is the approach to proving/disproving that they are equivalent under the set of spherical rotations?
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2answers
67 views

How many permutations of order $8$ in $S_{10}$?

Find the number of permutations of order $8$ in $S_{10}$. If $\sigma$ is one of them, find the number of subgroups in the subgroup generated by $\sigma$.
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Is the multiplicative group $\Bbb Z_{36}^\times$ cyclic?

I'm trying to answer this question but also understanding a smart method to find if a group like the one mentioned has a cyclic generator or not. I know that similar questions have already been asked ...
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2answers
41 views

An attempted proof of Cauchy's theorem for abelian groups using composition series.

I came up with this proof for the abelian version of Cauchy's Theorem (if a prime $p$ divides the order of an abelian group then it has a subgroup of order $p$). I'm hoping someone could please check ...
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1answer
39 views

$x$ generator of $(\mathbb{Z}/p^2\mathbb{Z})^* \implies$ $x$ generator of $(\mathbb{Z}/p^n\mathbb{Z})^*$

As it's indicated in the title, I have to show that $x$ generator of $(\mathbb{Z}/p^2\mathbb{Z})^* \implies$ $x$ generator of $(\mathbb{Z}/p^n\mathbb{Z})^*$, $p \neq 2$. I'm not sure of my proof,...
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1answer
77 views

Image of polynomial in $F[x]$ under derivation

Let $F$ be a field of characteristic zero and let $D$ be the formal polynomial differentiation map so that $$D(a_0+a_1x+a_2x^2+....+a_nx^n)=a_1+2a_2x+3a_3x^2+....+na_nx^{n-1}$$ Find the image of $F[x]$...
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3answers
139 views

$U\times W \cong V\times W $ does this imply that $U\cong V$

Let $U, V,$ and $W$ be smooth manifolds such that $U\times W$ is diffeomorphic to $V\times W $ does that imply that $U$ is diffeomorphic to $V$.? Is this result also true/False in other ...
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0answers
29 views

When does a simple Lie group contain a nonsimple subgroup?

The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $\text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well. The conformal group of ...
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82 views

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $ G\cong N \times G/N. $ [duplicate]

Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $$ G\cong N \times G/N. $$ I tried to prove this claim, but then it seems that since $G$ is abelian then every ...
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3answers
60 views

What are some applications of subdirect product?

I have studied direct products. I know a few applications of direct products, like group isomorphism, etc. What are some applications of sub-direct product of groups?
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1answer
56 views

Is there a non-simply connected subspace of $\mathbb R^2$ with trivial first homology?

Exactly what the title says. Can you find some topological space $X\subset\mathbb R^2$ such that $\pi_1(X)\neq0$, but $\mathrm H_1(X,\mathbb Z)=0$? I've been told that this paper shows that the ...
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2answers
67 views

Are all cyclic groups considered sub-groups of another group? [on hold]

Consider $Z_5^*$ as a group G, where the group operation is multiplication. Using 2 as a generator and element of G, a cyclic group can be generated: 2⋅0 (mod 5) = 0 2⋅1 (mod 5) = 2 2⋅2 (mod 5) = 4 2⋅...
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Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$.

Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
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41 views

Explain projective representation vs faithful representation

This is very basic. I am trying to explain projective representation vs faithful representation in a most naive way to a class of middle school students. My formal way of understanding is that ...
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3answers
59 views

How do cosets form a group?

Let G be a group and H be a Subgroup . Then if H is normal then the cosets form a group . I cannot understand what is meant by cosets form a group . To from a group , group operation on two elements ...
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1answer
40 views

Coloring of circle with 12 sections

We have a circle with 12 (equally sized) sections. In how many ways can you color the circle if we consider two results similar if you can get one from the other by rotating the circle? (I'm looking ...
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1answer
42 views

For a strong prime $p = kq+1$, $\langle g^k \rangle \le \mathbb{QR}_p$ [on hold]

We call strong prime to a prime $p$ of the form $p = kq+1$ with $q$ prime and $k \in \mathbb{N}$. Consider $\mathbb{Z}_p^{*} = \langle g \rangle$ and $\mathbb{QR}_p = \{x^2 \; mod \; p: x \in \mathbb{...
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2answers
57 views

Alternative proof that $U(n^2-1)$ is not cyclic for $n>2$.

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 4.84 ibid. and I want to solve it using only the tools available in the textbook so far. (A free copy of the book is ...
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1answer
38 views

Connection between Freiheitssatz and Magnus property

I am currently studying the Magnus property: Let $G$ be a group and $u, v \in G$. If the normal closures of $u$ and $v$ coincide, then $u$ is conjugate to $v$ or $v^{−1}$. I was told that this ...
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2answers
38 views

$P$ is a $p$-Sylow subgroup of $G$ and $H$ is a subgroup of $G$. Show that $P\cap H$ is contained in a $p$-Sylow subgroup of $H$

Let $G$ a group, let $H$ a subgroup of $G$ and let $P\in\operatorname{Syl}_p(G)$. Show that $P\cap H$ is a subset of some $U\in\operatorname{Syl}_p(H)$. I know that \begin{align} |G| &=p^a\cdot ...
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1answer
37 views

What is $Ext^1(\overline{\mathbb{Q}}^\times, \overline{\mathbb{Q}})$ in abelian groups?

I want to find a way to describe all the extensions of $\overline{\mathbb{Q}}^\times$ by $\overline{\mathbb{Q}}$, i.e., all the abelian groups $A$ (and the maps $\alpha$ and $\beta$) that fit into the ...
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56 views

subgroup of $GL(2,\mathbb{R})$, set of all invertible matrices over $\mathbb{R}$

Does $GL(2,\mathbb{R})$ contain a cyclic subgroup of order $5$, where $\mathbb{R}$ = set of all real numbers?
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1answer
48 views

If a prime $p$ divides the order of a group $G$ and $p^2 > |G|$, then there is a normal subgroup of order $p$ in $G$

I was wondering if there were a way to prove this without invoking the full force of the Sylow theorems. Here is my attempt: Suppose $G$ is a group, a $p$ prime divides its order, and $p^2 > |G|$. ...
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1answer
28 views

Find a homomorphism $f:\mathbb{Z}\rightarrow \mathbb{C^*}$ such that $\mathrm{im} f=\langle z \rangle$

Let $$z=\cos\frac{2\pi}{7}+i\sin\frac{2\pi}{7}$$ Find a homomorphism $f:\mathbb{Z}\rightarrow \mathbb{C^*}$ such that $\mathrm{im} f=\langle z \rangle$ Struggling to figure out how to start solving ...
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1answer
31 views

Is the inverse image of a group also a group for semigroup homomorphisms

If $\varphi : S \to T$ is a surjective semigroup homomorphism between semigroups and $G \subseteq T$ is a group, then is $\varphi^{-1}(G)$ also a group? I know that this result holds if $S$ and $T$ ...
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1answer
19 views

Deriving the expression $(J_{jk})_{ab}=-i(\delta_{ja}\delta_{kb}-\delta_{jb}\delta_{ka})$ for the $SO(4)$ generators?

I wish to derive the Lie algebra $so(4)$ of the generators $J_{jk}=-J_{kj}$ of ${\rm SO(4)}$ as $$[J_{jk},J_{lm}]=\delta_{jl}J_{km}+\delta_{km}J_{kl}-\delta_{jm}J_{kl}-\delta_{kl}J_{jm}.\tag{1}$$ ...
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1answer
61 views

Gallian's Exercise 4.76: a 2008 GRE practice exam question: “exactly two of $x^3$, $x^5$, and $x^9$ are equal, determine $\lvert x^{13}\rvert$”.

This question appears to be new here according to this and this. I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 4.76 ibid. and it's stated to be a problem from a 2008 GRE ...
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2answers
26 views

Verifying the calculation of the number of conjugate elements

I would like to verify my understanding: Consider action of $S_7$ on itself by conjugation. I'm trying to compute: $Stab_{S_7}((1 2))$ $Stab_{S_7}((1 2 3 4 5 6 7))$ $Stab_{S_7}((1 2 3)(4 5 6))$ I'm ...
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1answer
27 views

Showing that there exists an element in abelian group from Herstein

If $G$ is an abelian group where $a,b \in G$ have orders $m$ and $n$ respectively, I'm trying to show that there exists an element with order equal to $lcm(a,b)$. This is a verification of my proof, ...
1
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1answer
33 views

Equivalent form of the Whitehead problem

Let $M$ be an $R$-module and consider the following statement. $M$ is projective whenever the obvious group map $\tau: \text{Hom}_R(M, R) \to \text{Hom}_R(M, {R}/{I})$ is surjective for any ideal ...
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0answers
22 views

Right zero in a finite semigroup

Let $(M, \cdot)$ be a finite semigroup such that $$ x,y\in M\wedge \exists a,b\in M:x=a⋅y\wedge y=b⋅x\Rightarrow x=y. $$ Show that M contains at least one right absorbant element(or right zero). ...
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1answer
76 views
+100

$G$ with a central series that is different from the upper and the lower central series

Let $$1=G_0\leq G_1\leq ... \leq G_{n-1} \leq G_n = G$$ be a central series of the group $G$. That is, $G_{i-1}/G_i\leq Z(G/G_i)$ for all $i$. Let $$1=Z_0(G)\leq Z_1(G)\leq ... $$ $$... \leq \...