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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$A_n$ is generated by 3-cycles given $n\geq 3$. Is this proof correct?

The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$ In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the ...
2
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1answer
36 views

Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...
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22 views

Abelian subgroup of standard wreath product

Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows. Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with ...
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1answer
40 views

Quotient of direct sum of abelian groups

Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$? Many thanks in advance!
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1answer
29 views

How to find an element and a subgroup of a certain given order in $U(n)$

How can I find a subgroup of order $k$ in $U(n)$ or an element of order $k$? Here $U(n)$ is the group of units modulo $n$. For example, if $n=700$ and $k=6$ I know that since $700=5^2 \cdot 7 ...
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1answer
43 views

Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$? [duplicate]

Let $G$ be a finite group, $A\subseteq G$ and put $A^{-1}=\{ a^{-1}:a\in A\}$. Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?
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20 views

Smallest $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$

I solved the following exercise: Find an integer $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$. Here $U(n)$ is the group of units modulo $n$. To solve it ...
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1answer
73 views

If $f:\mathbb{Z} \to \mathbb{Z}$ is an isomorphism, prove that $f$ is the identity map. [on hold]

I am a little baffled by this question. Is it safe to assume that since $f$ is an isomorphism, $f (1) = 1$ ? And, if it is safe to assume this, could I construct a proof by induction, by using the ...
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1answer
40 views

In $U(55)$ show $x \mapsto x^3$ is injective

I am stuck on showing that $\varphi : U(55) \to U(55) $ given by $x \mapsto x^3$ is an isomorphism. I already knwo that $\psi: U(n)\to U(n), \psi(x) = x^k$ is an isomorphism if and only if $\gcd(k,n) ...
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22 views

Direct Product and Quotient of Groups

Quick (and basic) group theory question: Say G, H, K some (Lie) groups, does it in general hold that $$ (G \times H)/H = G $$ and that $$ H = K\times G \to K = H / G $$ And if so, does it then ...
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31 views

If $G$ acts $k$-transitive and $k > 5$ and $G$ is neither alternating nor symmetric, then $(n-k)! \ge 2n$

The following is an exercise from D. Robinson: A Course in the Theory of Groups. Let $G$ be a $k$-transitive permutation group of degree $n$ which is neither alternating nor symmetric. Assume $k ...
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1answer
23 views

Can the galois group be the symmetric group, if the discriminant is a perfect square?

Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square. Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of ...
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22 views

Centralizer of $(1,2,3)(4,5,6,7,8) \in A_{11}$

Let $$\tau = (1,2,3)(4,5,6,7,8) \in A_{11},$$ where $A_{11}$ is the group of even permutations. Let $H$ be the centralizer of $\tau$. I can easily prove that $H$ is a subgroup of $A_{11}$. How do I ...
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Prove/Disprove : Every polynomial with prime degree and coefficients in $[-1,1]$ has galois-group $S_p$

Conjecture : Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ ...
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2answers
30 views

$G$ a finite group such that $x^2 = e$ for each $x$ implies $G \cong \mathbb{Z}_2 \times … \times \mathbb{Z}_2$ ($n$ factors)

Let $G$ be a finite group such that $x^2 = e$ for each $x \in G$. I know already that $G$ is abelian and that the order of $G$ is $|G| = 2^n$ for some $n \geq 0$. Now I wish to show that $$G \cong ...
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0answers
17 views

Evaluate the Projection Operator for this Irreducible Representation of Dihedral Group

I am trying to compute the projector for the Dihedral group of order 12 ($D_{12}=D_{2n}$) for a certain Irreducible Representation. The representation is two dimensional and so I need to caculate ...
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1answer
28 views

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

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
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1answer
29 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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0answers
22 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
20 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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2answers
43 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
40 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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33 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
27 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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0answers
9 views

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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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
35 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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0answers
17 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
50 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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2answers
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$U(n)^2$ 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
42 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
21 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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0answers
28 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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1answer
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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
29 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
34 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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0answers
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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
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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 ...
4
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1answer
31 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
66 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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0answers
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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0answers
8 views

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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0answers
25 views

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
74 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 ...