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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Proof of a theorem about cardinality of quotient groups

I am looking for a proof of the following proposition: If $H_1<G, H_2<G$ and $[G:H_1]$ and $[G:H_2]$ finite. Then $[G:H_1\cap H_2]$ is also finite Here's what I've tryed to do: $[G:H_1 \cap ...
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33 views

Construction of Permutation Group (bounded order) from a Set of Permutation

Given a set of permutation $S$. It has $|S|$ (The cardinality of $S$) elements. Consider a subset $A\subset S$. We can construct a group using $A$. One possible algorithm could be- We start ...
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$x^{20}=1$ for all $x\in U(100)$

Show that $x^{20}=1$ for all $x\in U(100)$. In the book that I'm reading, it says "Since $U(100)\thickapprox Z_2\oplus\,Z_{20}$ we see that $x^{20}=1$ for all $x$ in $U(100)$".I think maybe that ...
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21 views

Decomposition of semi-simple Lie algebras

Background: Let $G$ be a finite-dimensional Lie group, with Lie algebra $L(G)$. A subspace $I\subset L(G)$ which satisfies $[L(G),I]\subset I$ is called an ideal of $L(G)$. A non-abelian Lie algebra ...
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1answer
21 views

Reduction of a representation of the Symmetric Group $S_3$

I have this representation of $S_3$ obtained in the usual way $$\varrho\left(\sigma\right)e_i=e_{\sigma_i}$$. Being more explicit the representation is this one: ...
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27 views

Differences between realizations and representation of a group

I am studying an introduction to group representation theory on my relativity class' lecture notes. I've previously learned in other classes and also on the Wikipedia article that a representation $T$ ...
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34 views

A question about groups

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. $H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some ...
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62 views

If two parallel lines meet at infinity, then what is their angle? [duplicate]

Since lines that meet at some point have an angle. And if parallel lines meet at infinity, then that what is the angle of two parallel lines that meet at infinity?
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1answer
24 views

Homotopy group of the conformal group

I would like to know which are the first three homotopy groups of the conformal group SO(4,2): $$ \pi_n(SO(4,2))=? \quad n=1,2,3 $$
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1answer
29 views

In some basis for a vector space $V$, its matrix is diagonal.

Let $\phi: G \to \text{Aut}(V)$ be an irreducible representation of a finite group $G$, where in some basis for $V$, all matrices $\phi(g)$ have real entries. For this basis, is it true that ...
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42 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. ...
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34 views

$U(2^n) \thickapprox Z_2\oplus Z_{2^{n-2}}$

Show that $$U(2^n) \thickapprox Z_2\oplus Z_{2^{n-2}}\,\,\,\,\,\,,\text{for }n\geq3$$ Well I think this can be done with the help of this theorem Let $m=n_1n_2.\,.\,.\,.n_k$ where $\gcd(n_i,n_j)=1$ ...
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3answers
47 views

Why do we have the following implication if $\phi$ is injective

If $\phi: G \rightarrow H$ is a homorphism, and if $\phi$ is injective, why do we have the following: $\phi(g) = e_h \implies g=e_g$
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2answers
47 views

A group of order pq with a single subgroup of order p [on hold]

Given a group $G$ of order $pq$ (such that $p,q$ are primes and $p < q$) that have a single subgroup of order p (named $H$) prove that $\forall h \in H , g\in G : ghg^{-1} = h$
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23 views

Proof of a theorem regarding group homomorphisms and kernels

I am looking for a proof of the following theorem: "Let $H<G$ then $H\unlhd G$ $\iff$ there exists a group $K$ and a group homomorphism: $\phi : G \rightarrow K$ such that $ker(\phi) = H$ There ...
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2answers
51 views

“By cancellation” in group theory?

I don't quite understand what the following "by cancellation" means in group theory: http://www.calpoly.edu/~brichert/teaching/oldclass/w2003412/solutions/solutions10.pdf, p. 9. Thus we may ...
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3answers
24 views

For an action of $G$ on a set, every point of some orbit has the same stabilizer if and only if this stabilizer is a normal subgroup.

Here is my proof of the first part. (=>). Let $G$ act on a set $X$ and for some $x\in X$. In order to show that $G_x$ is a normal subgroup of $G$, we will show that for all $g \in G$ we have $gG_x = ...
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53 views

Is there a name for the class of infinite groups with no proper infinite quotients?

As the title suggests, I would like to know if there is there a name for the class of infinite groups with no proper infinite quotients? I came across this paper ...
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31 views

Hall $\pi$-subgroup of a minimal normal subgroup of $G$

Let $B$ be a minimal normal subgroup of $G$ and suppose that $H \in$ Hall$_\pi$($B$). Then $B = S_1 \times \dots \times S_n$ where $S_i$ are simple groups. I'm not sure how $H = \langle H \cap S_i ...
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1answer
37 views

$U(st)$ is isomorphic to $U(s)\oplus U(t)$ where $s$ and $t$ are relatively prime.

Suppose $s$ and $t$ are relatively prime.Show that $U(st)$ is isomorphic to $U(s)\oplus U(t)$. I want to show that $\phi$ :$U(st)$ $\rightarrow$ $U(s)\oplus U(t)$ defined by $\phi(x)=(x\bmod s,x\bmod ...
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37 views

Pairwise distinct subsequence

Suppose there is an infinite sequence $S_n = (s_1, s_2, \dots )$ generated by a finite set of numbers $\{1, 2, \dots, n\}$. Given a number $m$ such that $m < n$, the subsequence $(s_i, s_{i+1}, ...
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2answers
87 views

Automorphism group of a graph = Automorphism group of that graph's adjacency matrix?

Is automorphism group (or set) of a graph $G$ equal to the automorphism group (or set) of adjacency matrix of $G$? Example: $G_1, G_2$ are separate graphs where $G_1^{\pi}= G_2$ and $ G= \bar ...
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1answer
39 views

How to detect automorphism of union of graphs?

On page 1 of Lecture 2, Algebra and Computation , (Course Instructor: V. Arvind), there is a theorem- Theorem 2. With Graph − Iso (graph isomorphism) as an oracle, there is a polynomial time ...
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1answer
80 views

Describe, as a direct sum of cyclic groups, given a map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$

I'm trying to resolve the next one: Describe, as a direct sum of cyclic groups, the cokernel of the map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$ given by left multiplication by the ...
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1answer
31 views

How do I know the fundamental group of an infinite graph is well defined?

I get that given a choice of spanning tree and base point for a (connected) graph, I can effectively change the base point through path conjugation, so there's no problem there. For finite graphs, the ...
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1answer
56 views

Possible mistake in Apostol's book: “An introduction to analytical number theory” (?)

On page 132 of Apostol's "An introduction to analytical number theory" : Theorem 6.6: Let $G'$ be a subgroup of a finite abelian group $G$, where $G' \neq G$. Choose an element $a \in G$, $a \notin ...
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1answer
32 views

Multiple correct question based on permutation group [duplicate]

I try to solve this the number of permutation in $S_{n}$ for $n\geq 4$ which are product of two disjoint $2$-cycles is $\frac{n(n-1)(n-2)(n-3)}{8}$. So after putting $n=5$ and $n=4$ I get different ...
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1answer
54 views

Is $SO(n)$ actuallly the same as $O(n)$?

$SO(n)$ is defined to be a subgroup of $O(n)$ whose determinant is equal to 1. In fact, the orthogonality of the elements of $O(n)$ demands that all of its members to have determinant of either $1$ or ...
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Show that $\varphi_{x,y}$ is a automorphism iff $\mathrm{ord}(x)=\mathrm{ord}(y)=n$ and $\langle x\rangle\cap\langle y\rangle=\{\hat{0},\hat{0}\}$.

Let $n\in\mathbb{N}$ and $$\varphi_{x,y}:\left(\mathbb{Z}/n\mathbb{Z},+\right)\to\left(\mathbb{Z}/n\mathbb{Z},+\right)$$ $$(\hat{m},\hat{l})\to m\cdot x+l\cdot y$$ a homomorphism. Show that ...
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1answer
45 views

Finite Almost Simple Groups

I want to study finite almost simple groups but I am not sure which would be the best texts to look at. Can someone please refer me to some books that teach the theory of finite almost simple groups?
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Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?

Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...
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30 views

Are these subgroups isomorphics? [closed]

Let $H_1\leq G_1$ and $H_2\leq G_2$. If $G_1\simeq G_2$ and $G_1/H_1\simeq G_2/H_2$ is $H_1\simeq H_2$ thanks
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Faithful irreducible character and Sylow subgroup

I am trying to solve the (very nice) exercise 5.25 from Isaacs, character theory. Assume that every Sylow subgroup of $G$ has a faithful irreducible character. Show that $G$ has one also. The ...
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Kazhdan's property (T) vs residual finiteness

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $(F)$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with ...
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1answer
54 views

Group Theory False Stements [closed]

Assume that $H_i$ is a normal subgroup of $G_i$ for $i=1,2$. Find an explicit counter example to the below false statements. a.) If $G_1\simeq G_2$ and $H_1\simeq H_2$, then $G_1/H_1\simeq G_2/H_2$ ...
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1answer
28 views

isomorphic subgroups of the additive group of rational numbers [closed]

Let $H$ and $K$ two subgroups of the additive group of rational numbers $( \mathbb{Q},+,-,0)$. Show that if there are positive integers $m$ and $n$ such that $mH \subset K$ and $nK \subset H$ then ...
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What does a maximal torus in GSpin$_{2n}$ look like?

I am interested in spin groups for a project at my university, and I was wondering: what would a maximal torus in GSpin$_{2n}$ look like, and how does one come to it? Does anyone maybe have a ...
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2answers
32 views

Suppose G is a group of order 4 and $x^2=e$ for all $x$ in G. Prove that G is isomorphic to $Z_2\oplus Z_2$. [duplicate]

Suppose $G$ is a group of order $4$ and $x^2=e$ for all $x$ in $G$. Prove that $G$ is isomorphic to $Z_2\oplus Z_2$. My attempt: (1) Show that $G$ is abelian. \begin{align*} \text{Take }x,y\in ...
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1answer
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Question on abelian group, does $G/H$ abelian $\iff [G,G]\leq H$.

Let $G$ a group and $H\lhd G$ a normal subgroup. I have a theorem that say that if $G/H$ is abelian, then $[G,G]\leq H$. I wondered if the converse hold, does it ? I recall that ...
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Character and centralizer of a subgroup

I am trying to solve the exercise 2.15 from Isaacs Character Theory. Let $\chi\in Irr(G)$ be faithful and let $H$ be a non-trivial proper subgroup of $G$ such that $\chi_{H} \in Irr(H)$. Show that ...
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1answer
21 views

Normalizer of a quotient

Suppose that $G$ is a group and $A, B$ and $H$ are subgroups of $G$ with $B\unlhd A$. In a paper I have read, they say that the normalizer of $A/B$ in $H$ is $N_H(A/B) = N_H(A) \cap N_H(B)$. I am ...
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84 views

Number of Subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime

As the title says, I am interested in finding all subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime. We did not cover the Sylow-theorem so far in the lecture. What I noticed so far: As ...
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2answers
69 views

Is this notation on the restriction of a function in group theory common?

If $f: X \rightarrow Y$ is a function between sets $X$ and $Y$, then a common notation to use when we want to restrict $f$ to a certain domain $X' \subset X$ is $f|_{X'}: X' \rightarrow Y$. I'm doing ...
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1answer
19 views

Calculating the group action of $A_4$ on the rotational symmetries of the cube.

The following question is taken from M.A. Armstrong's Groups & Symmetry #17.3. It reads as follows: Having identified $S_4$ with the rotational symmetry group of the principal diagonals of the ...
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27 views

Is there a non-solvable amenable torsion-free countable ICC group?

Question: Is there a non-solvable amenable torsion-free countable ICC group? If so, is there a classification program of such groups? Remark: In the torsion case, there is the finitary symmetric ...
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1answer
32 views

The Rotation Group of a Cube

Show that the group of rotations of a cube is isomorphic to $S_4.$This proof is from Gallian's Abstact Algebra Theorem $7.3$ Proof:Using the Orbit-Stabilizer Theorem we know that the group of ...
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1answer
76 views

On the converse of Schur's Lemma

Let $G$ be a finite group and $F$ a field with $\mathrm{char}(F)=0$ or coprime to $|G|$. Let $V$ be a $FG$-module in a way that every $ FG$-homomorphism $ f : V \to V $ is given by $f(x)= \lambda x ...
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5answers
59 views

Proving that the groups $S_3$ and $D_6$ are isomorphic [duplicate]

In particular, $S_3$ is the group of permutations of $\{1,2,3\}$, and $D_6$ is the dihedral group of symmetries of the triangle (written as $D_{2\cdot 3}$). In generator-relation form, $D_6 = ...
3
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1answer
49 views

Free group uniqueness from the universal property

Just a note, I'm using Dummit and Foote (specificially second edition Prentice Hall, section 6.3 "A word on free groups") as my reference material. I also found a similar question here, but this is ...
2
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1answer
34 views

Finding normal subgroups

Let $G=\langle e, r,..., r^{n-1},s,sr,...,sr^{n-1} \rangle $ be a dihedral group with $2n$ elements, for $ 3 \leq n$. Prove that the only normal subgroups of $G$ are $\langle r^d \rangle$ (where $d$ ...