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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For which $p$ is $G_p = \{\sigma \in S_{12}: ord(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: ord(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to consider $p \le ...
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22 views

If $a$ is the only element of order $2$ in a group, show that $a$ is in the center of the group.

If $a$ is the only element with order $2$ in a group $G$, then $a \in Z(G)$. I'm studying for a test and I can't figure out how to prove it. What kind of methods might I try to solve this ...
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21 views

Verification: A self-conjugate element in an odd-order finite group is the identity

I think I've found a proof of the following, but my proof is horrible, and I feel like I've made a mistake or that I've missed an important principle: Theorem: Let $G$ be a finite group of odd order ...
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1answer
17 views

Studying the symmetry group of the dodecahedron by introducing axes.

For my bachelor project I'm studying the symmetry of the Platonic solids (as a start at least). I computed the symmetry group of the tetrahedron by labeling the vertices, and the cube by labeling the ...
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42 views

Proving an Stabilizer is the whole group.

Let $G$ be a group of $143$ elements acting on a set $X$ of $108$ elements. I need to prove that there is one element whose Stabilizer is the whole group. I have tried doing it using the Orbits ...
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3answers
33 views

subgroup having index $2$ of $R^*$

The question is to find all the subgroups of $R^*$ (non-zero reals under multiplication) of index $2$. The index can be found out for finite groups. How to find subgroups having certain index for an ...
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31 views

Finding subgroups with a specific property.

I'm trying to find subgroups with the following properties: if $[G:H]=n$ there is a $g∈G$ so that $g^n≠e$. What do I do know is that $H$ cannot be normal (previous exercise). I just can't find any ...
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2answers
26 views

If $H \triangleleft K \leqslant G$, which requirements must be placed on $K$ in order to obtain $N \triangleleft G$?

Let $H$ be a normal subgroup of $K$, which is a subgroup of $G$. By just sketching some computations it seems that $H$ is not necessarily a normal subgroup of $G$ even if $K$ is a normal subgroup of ...
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the numbers Tr$(X^n)$ determine the conjugacy class of semisimple $X \in GL_n(\mathbb{C})$.

L.S., I am reading a paper of D. Blasius, where he states that the numbers Tr$(X^n) $ determine the conjugacy class of a semisimple element $X \in GL_n(\mathbb{C})$. I am having trouble proving this. ...
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28 views

Is $U(1)$ a normal subgroup of $U(2)$ and does the question even make sense?

I have been wondering whether $U(1)$, defined as the group of complex phases (edit for clarity: I mean complex numbers of unit absolute value, such as $e^{i\alpha}$ with $\alpha \in \mathbb{R}$) with ...
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3answers
29 views

Question regarding the normality of a certain subgroup of a group

Let $G$ be a group and $N$ a normal subgroup of $G$. Let $H=\{g\in G\mid gn=ng\space \forall n\in N\}$. Prove that $H$ is a normal subgroup of $G$. I've tried seeing if we can write $H$ as the ...
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15 views

$SU(n)$ generators

What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. ...
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1answer
45 views

Proper subgroup of $S_{15}$ that strictly contains $\sigma $

How does one prove that: There exists no cyclic proper subgroup of $S_{15}$ that strictly contains the following permutation (of order $10$)? $$\sigma = (1, 2,3, 4,5, 6, ...
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27 views

Intersection of subgroup with Sylow subgroup [on hold]

Let $p$ be a prime number. If $P$ is sylow $p$ subgroup of $G$ of some finite group $G$ then for every subgroup $H$ of $G$, $H \cap P$ is $p$ sylow subgroup of $H$? True of false?
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24 views

$U(n)^2$ is a proper subgroup of $U(n)$

I'm trying to show that $U(n)^2$ is a proper subgroup of $U(n)$. Here $$ U(n)^2 = \{x^2 \mid x \in U(n)\}$$ where $U(n)$ is the group of units modulo $n$. My idea was to argue as follows: Consider ...
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1answer
41 views

If a group has no subgroups other than the identity and itself, then it is finite and is of prime order [duplicate]

I want to prove that if a group has no subgroups other than the identity and itself, then the order of the group is a prime number. A hint would be appreciated. Is there any theorem on the relation ...
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1answer
24 views

Showing cyclic group

Is $\Bbb R_{+}$ under multiplication cyclic? Would it suffice to show that this group is isomorphic to a cyclic group? For example, $\Bbb Z_+$ is cyclic under addition, $\Bbb R_{+}$ is isomorphic to ...
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1answer
18 views

Proofs of congruence relations

Exercise 2.3 from "Introduction to Mathematical Cryptography" Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$ of order $r$. (a) Suppose that $x = a$ and $x = b$ are both integer ...
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4answers
41 views

Showing the group in $\Bbb R$

I have the following problem I am confused about: Let $x,y\in \Bbb R, x\ast y=xy+x+y$ Is $\Bbb R$ a group? I wrote $(x\ast y)\ast z= x\ast (y\ast z)$, then calculated it, associativity did not hold. ...
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24 views

2016 AMC 10A #18 — Number of ways to label vertices of acube

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for ...
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2answers
23 views

If $G$ is isomorphic to $H$, show ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$

For every $\alpha\in{\rm Aut}(G)$, I've defined $A:H\rightarrow H$ by $$A(h)=\phi(\alpha(\phi^{-1}(h)))$$ where $\phi$ is an isomorphism from G onto H, and I've shown that $A\in {\rm Aut}(H)$. What I ...
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1answer
14 views

product of torus and affine space

Given the set $\mathbb{C}^*\times \mathbb{C}$ with the group structure given by $(x,y)\cdot (z,w) =(xz, zy+wx) $. How can I check that this is reductive or not? ? I think that its maximal normal ...
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15 views

Definition of generators in the context of groups as languages

In the book "Word processing in groups" by Epstein et al. (p.28-29), the definition of generators begins with the following sentence: Let $G$ be a group, $A$ an alphabet and $p \colon A ...
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1answer
25 views

Proving that a group representation is *not* a direct sum of irreducible represenations.

Problem Statement: Let $x$ be a generator of a cyclic group $G$ of order $p$. Sending $x\mapsto \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}$ defines a matrix representation $G\rightarrow ...
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1answer
33 views

Subgroup H group G.

Let $G$ be a group, $H$ a subgroup of $G$, and $N:=\cap_{x\in G} \ \ x^{-1}Hx$. Prove, that $N$ is normal subgroup in $G$. I did this: Let $g\in G$. Whether $g(xhx^{-1})g^{-1}\in N$? Take $f=gx$. ...
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7 views

Why are two transitive actions of a group G equivalent if there exist an automorphism of G swapping two point stabilisers?

Let $G$ be a group acting transitively on two sets $\Omega_{1}$ and $\Omega_{2}$. Also let $w_{i}\in\Omega_{i}$ and suppose there exists $\alpha\in Aut(G)$ such that $\alpha(G_{w_{1}})=G_{w_{2}}$, ...
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0answers
17 views

Isomorphism of a quotient group [duplicate]

I have that $G=S_4$ and $N = \{1, (12)(34), (13)(24), (14)(23)\}$, and thus far I have shown that N is a normal subgroup of G. I'm trying to figure out what group $G/N$ will be isomorphic to, but I ...
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1answer
29 views

A set containing more than half elements of a group [duplicate]

I wish to prove the exercise which states that for a set $A$ containing more than half elements of a group $G$, every element of $G$ is a product of two elements of $A$. My attempt: By Lagrange ...
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1answer
19 views

If $K$ is a normal subgroup of $G$, $K$ has prime order, and $Z(G) = {1}$, then the Centralizer of $K$ in $G$ is equal to $K$

Prove that if $K$ is a normal subgroup of $G$, $K$ has order $q$, where $q$ is prime, $G$ has order $pq$ where $p$ is a prime and less than $q$, and $Z(G) = {1}$, then the Centralizer of $K$ in $G$ is ...
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2answers
65 views

Isomorphism of a group $G = \left\langle x , y \mid x^5 = y^2 = e , x^2y = yx \right\rangle$ [on hold]

Let $G$ be a group whose representation is $G = \left\langle x , y \mid x^5 = y^2 = e , x^2y = yx \right\rangle$ . Then $G$ is isomorphic to a) $\mathbb Z_5$ b) $\mathbb Z_2$ c) $\mathbb Z_{10}$ ...
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19 views

Order of a certain finitely generated group

Suppose I am looking at the group $W=\langle s_\alpha, t_\beta \rangle$ where $s_\alpha$ and $t_\beta$ are reflections in $\mathbb{R}^2$ coming from two vectors $\alpha$ and $\beta$ making an angle of ...
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Generating space representation of all elements of a point group using generators

A well known set of groups are the 32 three-dimensional crystallographic point groups which elements represent the transformation matrices of the symmetry elements (rotation, reflection etc.). If one ...
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Suppose that $G$ is the disjoint union $G=\cup_{i=1}^n Sg_iT$.Prove that $[G:T]=\sum_{i=1}^n[S:S\cap g_iTg_i^{-1}]$ [on hold]

Let $S,T\leq G$,where G is a finite group, and suppose that $G$ is the disjoint union $$G=\cup_{i=1}^n Sg_iT$$ Prove that $[G:T]=\sum_{i=1}^n[S:S\cap g_iTg_i^{-1}]$ I don't have any idea how to start ...
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6answers
71 views

Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n

The question is given in the title. I am unable to come up with a counterexample and I'm thinking this could apply to cyclic groups but not necessarily to general groups. Does anyone have any ideas or ...
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1answer
23 views

$G$ is a finite group, if $ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$

$G$ is a finite group of order $n$, then if $a,b\in G : ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$. multiply both sides by $ab^{-1}$ we get $(ab^{-1})^2 = ab^{-1}ab^{-1}=ba^{-1}ab^{-1}=1$ so ...
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1answer
38 views

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R\{1}. [on hold]

For a, b ∈ R we define a ∗ b := ab − a − b + 2 ∈ R. Furthermore let G := R \ {1}. (a) Show that a ∗ b ∈ G for all a, b ∈ G. (b) Show that G together with the binary operation G × G → G, (a, b) |→ a ...
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1answer
24 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
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Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$.

Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$. Let $S_1,S_2\leq G,g\in G$ What i had done, $x\in (S_1 \cap S_2)g$. Then $x=sg,s\in S_1\cap S_2$. So clearly, $x\in ...
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Can we define structures like groups or monoids in the context of pure category theory?

In a category $\mathcal C$ with terminal object $1$ and objects $A$, $B$, $C$ we have $\quad$ $A\times (B \times C) \cong (A\times B) \times C$; $\quad$ $1 \times A \cong A \cong A\times 1$; ...
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Find all homomorphisms $Q \rightarrow \mathbb{Z}_8$

Let $Q$ be the quaternion group. Find all homomorphisms $\phi: Q \rightarrow \mathbb{Z}_8$ What I get into is one big ifology: Of course $\phi(1) = 0$, then $0 = \phi(1) = \phi(-1 \cdot (-1)) ...
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Structure of frobenius groups.

Definition- A groups $G$ is called Frobenius if it has a proper nontrivial subgroup $H$ such that $H \cap H^g=1\ \forall\ g\in G-H$. Do we have a structure or classification theorem for (finite) ...
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1answer
29 views

What am I missing here: $U(144) \neq U(140)$

I'm confused about the following exercise: Prove that $U(144)$ is isomorphic to $U(140)$. Here are my thoughts: $$U(144) = U(12^2) = U(3^2)\oplus U(2^4) = \mathbb Z_{6} \oplus \mathbb Z_{8}$$ and ...
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1answer
22 views

Why $U(p^m) \oplus U(q^n)$ is not cyclic

I tried to solve the following exercise: Let $p,q$ be odd primes and $n,m$ positive integers. Explain why $U(p^m) \oplus U(q^n)$ is not cyclic. I solved the question as follows: We have $U(p^m) ...
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25 views

Number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$

I tried to determine the number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$. Please could someone tell me if this is correct? $$ \text{Aut}(\mathbb Z_{720}) \cong U(720) \cong U(9) ...
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1answer
38 views

How to Identify a Quotient of a Given Free Group

$\newcommand{\Z}{\mathbf Z}$ Problem. Let $G$ be the free group generated by three symbols $a, b$ and $c$, and let denote $G$ by writing $F(a, b, c)$. Let $N$ be the normal subgroup of $G$ ...
2
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1answer
21 views

Action via automorphism

I want to ask what does it mean to say a group $A$ acts on $N$ via automorphisms. It is a notion used in M.Isaacs book and I am not familiar with. I tried to find how it is defined but a scanned e ...
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1answer
51 views

Group of order 6 contains an element of of order 3

I need to show that if $G$ is non-abelian group of order $6$ then it contains an element of order $3$. I don't know how to proceed. Any kind of help/hint is appreciated.
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1answer
42 views

Determine all the subgroups of the dihedral group $D_{15}$

Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup? Not sure where to start with finding all the subgroups of $D_{15}$ ...
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1answer
40 views

showing non-isomorphism of groups [duplicate]

How do I prove that there is no isomorphism between $\Bbb Z$ under addition and $\Bbb Q$ under addition? They both are infinite order. I thought they might be isomorphic. Help would be ...
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1answer
43 views

Groups isomorphic to $S_{4}/N$

Let $G = S_4$ be a group, $N = \{1, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ a normal subgroup of G. It's easy to see that $G/N$, the set of cosets is $G/N = \{a, b, c\}$, where $$a = \{(1), (1, ...