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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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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78 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
40 views

How is a characteristic subgroup verified?

I know the definition of a characteristic subgroup: $\sigma (H)=H$ for all $\sigma \in \text{aut}\, G$ where $H \leq G$. But, I do not understand how $\sigma$ is defined. Surely we can map $H$ to $H' ...
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40 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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2answers
19 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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3answers
40 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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38 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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45 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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12 views

Prove that $2\mathbb{Z} \otimes A$ is cyclic group order $2$ [on hold]

$A$ is cyclic group order $2$. Prove that $2\mathbb{Z} \otimes A$ is cyclic group order $2$ ?
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1answer
27 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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44 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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32 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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1answer
36 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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176 views

Isomorphism in certain groups.

Let $G = \langle \Bbb Z^3_2,+\rangle $, where as $ \Bbb Z_2= \{0,1\} $, $ \Bbb Z^3_2 = \{(x,y,z)\mid x,y,z \in \{0,1\}\} $, and the operation in $ \Bbb Z^3_2 $ is defined by $ (x_1,y_1,z_1) + ...
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2answers
529 views

Conjugacy classes in $ D_4$

Let G be group of all symmetries of square. Find number of conjugate classes in G. I tried this question just as we do for $S_n$ that the number of conjugate classes in $S_n $ is partition number of ...
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1answer
67 views

Character table of $S_3 \times C_2$

I need get of character table of $S_3 \times C_2$. How to make this character table? The representation is a $\psi (g,h) = \rho_1 (g) \rho_2 (h)$ with $\deg (\rho _2) = 1$ and $\rho _1 $ ...
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1answer
53 views

Let a group G act on a set $X$, and suppose that $x,y \in X$ lie in the same orbit. Prove that $G_y=g^{-1}G_xg$ for some $g \in G$

Let a group G act on a set $X$, and suppose that $x,y \in X$ lie in the same orbit. Prove that $G_y=g^{-1}G_xg$ for some $g \in G$ Ok, lets assume $G$ acts on $X$,where $x,y \in X, g \in G$ and ...
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30 views

Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd S_\mathbb{N}$. [on hold]

Let $S_\infty \subset S_\mathbb{N}$ be the set of permutations of $\mathbb{N}$ which are the identity on all but a finite number of elements. Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd ...
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A4 has no subgroup of order 6

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = ...
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3answers
18 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.

My logic is flawed in the first part and I have no idea what to do in the second. (=>). Let $G$ act on a set $X$ and for some $x\in X$ let $y \in G(x)$, the orbit of $X$. From our assumption $G_x = ...
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4answers
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If there are injective homomorphisms between two groups in both directions, are they isomorphic?

If I have two groups, $G$ and $H$ and two injective homomorphisms $\phi:G \to H$ and $\psi: H \to G$, then by the first isomorphism theorem applied to $\phi$, we have that $G \cong \mathrm{Im} ...
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29 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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35 views

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
32 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 ...
3
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0answers
34 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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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
38 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
434 views

On group automorphism of subgroup a group $G$

Let $G$ be a group and $H$ be a subgroup of $G$. When is $\rm{Aut}(H)$ a subgroup of $\rm{Aut}(G)$?
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1answer
34 views

Study of specific Quotients of a $p$-group in MAGMA

In Magma, I wanted to do a program, but since I was getting errors in "basic" commands, I am in trouble. Here I will describe a problem to be done in MAGMA, then its interpretation in MAGMA. I would ...
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1answer
55 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
30 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
43 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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67 views

Count the permutations which are products of exactly two disjoint cycles.

Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$ such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$. Calculating $a_4$: Possible ...
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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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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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4answers
2k views

Showing non-cyclic group with $p^2$ elements is Abelian

I have a non-cyclic group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, ...
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3answers
79 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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1answer
49 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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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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2answers
54 views

Prove that if $G$ is cyclic and infinite then $G$ is isomorphic to $\mathbb{Z}$

Assume $G$ is generated by $a$, so $G = \langle a\rangle$. Since $G$ is infinite for all $m \in \mathbb{Z}$, $a^m \neq e$. Suppose $a^h = a^k$ then $a^h\cdot a^{-k} = a^{h-k} = e$, but this is a ...
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26 views

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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4answers
28 views

Are these subgroups isomorphics? [on hold]

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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41 views

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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1answer
201 views

What does it mean for the group of rotation matrices to have a “manifold structure”

I have just been exposed to rotation matrices, and it is showing extremely strong connection with group theory and differential geometry which I am both totally inept at. Can someone in simple term ...
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1answer
599 views

General linear group/special linear group is isomorphic to R*

Let $GL(n,\mathbb{R})$ be the group of invertible $n \times n$ real matrices, let $SL(n,\mathbb{R})$ be the group of $n \times n$ real matrices of determinant $1$, and $\mathbb{R}^*$ be the group of ...
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1answer
40 views

Group Theory False Stements [on hold]

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
27 views

isomorphic subgroups of the additive group of rational numbers [on hold]

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 ...
5
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1answer
37 views

Showing normalizer of Galois group

Let $E/F$ be a Galois extension, and let $B$ be an intermediate field between $E$ and $F$. Let $H$ be the subgroup of $Gal(E/F)$ that maps $B$ into itself (but does not necessarily fix $B$). Prove ...
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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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0answers
11 views

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 ...