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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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Spin 1/2 representations of Lorentz group

For spin 1/2 we have the two representations of the Lorentz group given by: $$M_1 = e^{i(\vec{\theta} - i\vec{\phi})\vec{\sigma}/2}, \quad M_2 = e^{i(\vec{\theta} + i\vec{\phi})\vec{\sigma}/2} \tag1$...
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Examples of Polycyclic Group

I'm reading about polycyclic groups recently. Could anyone please give me some example of polycyclic groups?
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Cayley graph of Rubik's cube group

(a) I would like to know whether there is a group theoretic approach for calculating the diameter of the Cayley graph of Rubik's Cube group. I know it's been proved that the above diameter is $20$ ...
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Is $H/L$ isomorphic to $K^*\times K^*$ where $K^*=(K_{\ne0},\cdot)$ with $K$ field and $H, L$ certain subgroups of $GL(2,K)$?

Let $K$ be a field, $H=\left\{\begin{bmatrix}a&b\\0&d\end{bmatrix}:a,b,d\in K, ad\ne0\right\}$, $L=\left\{\begin{bmatrix}1&b\\0&1\end{bmatrix}:b\in K\right\}.$ I'm asked to prove that ...
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Free groups: Finding words vanishing in two different situations

Let $\varphi_i:G\to H_i$ for $i=1,2$ be two group homomorphisms. I want to find elements in $\mathrm{Kern}(\varphi_1)\cap \mathrm{Kern}(\varphi_2)$ which are not contained in the commutator $[G,G]$. ...
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1answer
25 views

Exercises on character group $G^*$ of a group $G$

Let $S^1=\{z \in \mathbb{C} : |z|=1\}$, which is a group under multiplication of functions. For a group $G$ define $G^*=Hom(G,S^1)$, the character group of $G$. Prove that $G^*$ is indeed a group ...
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1answer
46 views

I think these groups don't exists

A group $G$ is said to be decomposable if $G= A \times B$ ( direct product ). I am looking for the example of indecomposable groups (non-abelian) with very large center relative to the order of group. ...
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1answer
18 views

Sylow's First Theorem acting on Abelian Group

Background In the book of Judson's book on abstract algebra, Sylow's First Theorem is proved by first invoking the class equation and then considering the case where $p$ can/cannot divide $[G:C_G(g)]$...
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Presentation of groups affine transformation $x\to x+1$ and $x\to 2x$ is Baumslag-Solitar group $BS(1,2)$ [on hold]

Let $T:x\to x+1$ and $D:x\to 2x$ generated $x$ generated the group Aff$_{+}\mathbb{Z}([\frac{1}{2}])$ of affine transformations of the ring $\mathbb{Z}([\frac{1}{2}])$. Why This group has the ...
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1answer
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Determine all homomorphisms from $Q$ to $Q_{>0}^\times$.

This question is from a past year paper: Determine all homomorphisms from $Q$ to $Q_{>0}^\times$. Let $Q$ denote the group of rationals under addition. Let $Q_{>0}^\times$ denote the group of ...
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multiplicative abelian linearly ordered group on $\Bbb R-\{0\}$

It seems that $\Bbb R\setminus\{0\}$ does not create an alo-group (abelian linearly ordered group) with the "normal" order on $\Bbb R$. Is it possible to define any other order (including a trivial ...
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1answer
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if H is f.g, $Q\lhd_f H$ and $\phi \in \operatorname{Aut}(H)$ then $\bigcap\limits_{i \in \mathbb Z} \phi^i(Q) \lhd_f H$

I need to prove that if H is finitely generated, $Q\lhd_f H$ (normal of finite index) and $\phi \in \operatorname{Aut}(H)$ then $\cap_{i \in \mathbb Z} \phi^i(Q) \lhd_f H$. Given that H is f.g and Q ...
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1answer
22 views

Principal bundles with quotient map

I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $\pi: G \rightarrow G/H$, then $(G, \pi, G/H, H)$ is a $H$-principal bundle ...
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A relation between self-paired orbitals of a group action and its associated permutation representation

Let $G$ be a finite group. Suppose $G$ acts on a finite set $X$. Consider the permutation representaion or character associated with the action, call it $\pi$. Since permutation character is the ...
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1answer
22 views

Intermediary Extensions of $\mathbb{K}=\mathbb{Q}(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})$.

First, I must prove this extensions is Galois, by some algebra I proved that: $\mathbb{Q}(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})=\mathbb{Q}(\sqrt{2},i\sqrt{2})=\mathbb{Q}(\sqrt{2},i)$. And since ...
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1answer
27 views

a normal k-1 transitive subgroup

I want to show that if G is k-transitive and N is a normal nontrivial subgroup then is K-1 transitive. I know that I should use the fact G preserve the orbit of N's action of X. so for some Y on N's ...
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1answer
36 views

Proof of third isomorphism theorem [duplicate]

Let $G$ be a group, $H<G$ a subgroup of $G$ and $K<H$ a subgroup of $H$. Assume that $[G:K]$ is finite. Then: $[G:K]=[G:H][H:K]$. In the case where $G$ is finite, the proof is pretty simple ...
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23 views

Isotropy/little group of $O(n)$

I'm trying to prove that the little group of $O(n)$ acting on a $k$-dimensional subspace of $\mathbb{R}^n$, call it $V$, is $O(k)\times O(n - k)$ due to the Grassmann manifold is isomorphic to $O(n)/(...
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1answer
43 views

Equivalent conditions for a Group $G$ with order $p^2q$ ( with $p>q$ both prime) be abelian.

I saw this homework many times, but always asks in the statament that $p^2 \not\equiv 1$ (mod $q$) and $q \not\equiv 1$ (mod $p$). But today in a book text of Galois theory I Saw a similar example ...
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Show $A_4 \cong \langle a,b,c|a^2=b^3=c^3=abc=e\rangle$ (The von Dyck group(2,3,3)) [on hold]

Show $A_4 \cong \langle a,b,c|a^2=b^3=c^3=abc=e\rangle$. Here von Dyck group is $D(n,m,l):=\langle a,b|a^n=b^m=(ab)^l=1 \rangle$, it is the subgroup of index 2 in the triangle group $\triangle(n,m,l) :...
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1answer
36 views

Question on alternating groups [on hold]

An alternating group is the group of even permutations of a finite set. Question : Is there any theorem like $G$ is an alternating group iff something.. I tried on internet, but did not get anything....
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why is AxA Transitive, Reflexive, and Symmetric [on hold]

I just started group-theory. and we have this question but the professor never explained this material, I have no idea how to prove it. the question as asked by the professor(and Im translating it to ...
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2answers
28 views

Group of order 275 act on set of size 18, what is the minimum number of orbit of lenght 1?

Let $G$ be a group of order 275 acting on set of size 18, what is the minimum number of orbit of length 1? I think it is 2 because we then have $1+1+5+11$ all of the numbers in the sum are divisors ...
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19 views

Group with integer relation [on hold]

Let $G$ be an abelian group generated by some $x_1,\dots, x_n$ and a relation $\alpha_1x_1+\dots\alpha_nx_n=0$ with $\alpha_i\in\mathbb{Z}$. Is there a way to simplify this relation to $\alpha=gcd(\...
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1answer
26 views

on such group whose inner automorphisms group isomorphic to S3. [duplicate]

Let $\frac{G}{Z(G)}≅S_3$, such that $S_3$ is permutation group on 3 letters and $Z(G)$ is non trivial central subgroup of $G$. What are the possibility of group $G$ and does there exist always an non-...
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1answer
34 views

Relationship between prime power and the divisor

Background When trying to prove that a group G has a non-trivial centre if its order is a power of a prime $p$, there involves a step in which we claim the number of left cosets of a centraliser ...
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Subsemigroup of a finite semigroup

Let $S$ be a finite semigroup and $T \subseteq S$ which satisfy the following property: For $x, y \in T$, we have $x, y \in \langle z \rangle$ for some $z \in S$. If $H \subseteq S$ satisfy the above ...
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1answer
39 views

Order of taking quotients?

Let $G$ be a group and $M,N$ be its normal subgroups. I am thinking about the relationship between $(G/M)/N$, $(G/N)/M$, $G/(M/N)$ and $(G/M\cup N)/(M\cap N)$ (they are not proper notions but I am ...
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1answer
34 views

Real Grassmann manifold and orthonormal groups

I'm trying to prove that the Grassmann manifold $$G_k(\mathbb{R}^n) = \{E = {\rm {\it k} - dimensional\ subspace\ of\ } \mathbb{R}^n\}$$ is equivalent to: $$G_k(\mathbb{R}^n) = \frac{O(n)}{O(k)\...
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How to decide structure of quotient group by a concrete matrix?

Let $A$ be a $nxn$ matrix with integral coeffecients and nonzero determinant, I know that we can compute order of the finite abelian group $X=coker(\Bbb Z^n \overset{A}\rightarrow \Bbb Z^n)$ by just ...
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1answer
25 views

Definitions of normalisers for infinite groups

If G is a group and A is a subset of G, the normaliser of A in G can be defined as either (1) $N_G(A) = \{g \in G\ |\ gag^{-1} \in A, \forall a \in A\}$ (2) $N_G(A) = \{g \in G\ |\ gAg^{-1} = A \}$ ...
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“Intersection” of quotient groups, modulo $\mathbb{Z}^n$

Let's say that $M,N$ are two invertible $n\times n$ integer matrices, such that all their eigenvalues are greater than $1$ in magnitudes. Then we define $$K_M:=(M^{-1}\mathbb{Z}^d)\cap[0,1)^n,\quad ...
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Show $Gl_3(\mathbb{F}_5)$ has a normal subgroup of index 4

Show $Gl_3(\mathbb{F}_5)$ has a normal subgroup of index 4. Here is my attempt: $|Gl_3(\mathbb{F}_5)|=(5^3-1)(5^3-5)(5^3-5^2)=124*120*100=1488000=2^7*3*5^3*31$. So I'm stuck.
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1answer
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Asymptotic behaviour of the sequence of the number of groups of order $n$

Let $f(n) = \text{# of groups of order} \ n$. I want to study the asymptotic behaviour of this sequence as $n \to \infty$. Clearly $\lim \inf f(n) = 1$ and $\lim \sup f(n) = \infty$, so the sequence ...
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Normal Subgroup of G Fixes All Elements of Set if G Acts Transitively?

I am trying to prove the following equivalency: $H$ is normal in $G$ $\iff$ for any set $\Omega$ on which $G$ acts transitively, if $H$ fixes some $x \in \Omega$ then $H$ fixes every element in $\...
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1answer
32 views

number of orbits of $A_5$ acting by left multiplication in $S_5$

Looking for a very fast/"smart" way to compute this number (it was a question asked on an hour-long exam I recently took, so listing everything out for each element in $S_5$ was not an option since I ...
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Product of complete sets of representatives of the cosets

Suppose we have the subgroups $H \subseteq K \subseteq G$ (not necessarily normal), all finite groups. Denote $\mathcal{R}^G_H$ be a complete set of representatives of the cosets of $G$ in $H$, and ...
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1answer
44 views

Finding idempotents in group algebra over $A_n$

Let $G=A_4$ be the alternating group on 4 letters, and let $R = \mathbb{C}[G]$. Then $$\mathbb{C}[G] = U\oplus U' \oplus U'' \oplus V^{\oplus 3},$$ where $U,U',U''$ are the three 1-dimensional ...
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Weight diagram for $(2 \, 1)$ representation of $SU(3)$

Decided to practice my knowledge of representation theory by constructing the weight diagram for the representation $(2 \, 1)$ of $SU(3)$. This is apparently the $\mathbf{15}$, but when I use what I ...
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1answer
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Setwise stabilizer and relationship with pointwise stabilizer

Let $ T\subseteq S$ denote a subset of $S$ and $G$ denote the group acting on $S$. Let $$G_T=\bigcap_{s\in T} G_s \quad\text{and}\quad G_{\{T\}}=\{ g\in G: T = T^g \} $$ represent the pointwise ...
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1answer
34 views

How to construct a subgroup.

Consider the 2×2 matrices that (with respect to the standard basis of R2) represent rotation around the origin over 120 degrees and reflection in the x-axis. Construct the smallest possible subgroup ...
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Solvable group $G$ isn't a cyclic $p$-group

Let $G$ be a nontrivial finite solvable group. We want to show that $G$ has proper subgroups like $A$ and $B$ such that $G=AB$ iff $G$ isn't a cyclic $p$-group. I use the below fact to solve the ...
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1answer
24 views

Flag in vector space $V$ and a group $G$ stabilising it. Why is $G$ solvable?

Let $V$ be a $k$-vector space of dimension $n$. Take a flag $$0 = V_0 \subsetneq V_1 ...\subsetneq V_{m-1} \subsetneq V_{m} = V , $$ and a subgroup $G \leq GL(V)$ which stabilizes this flag, so ...
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1answer
48 views

Let $H$ a subgroup of $G$. $\exists g \in G$, $g \notin H$, such that $gH=Hg$, and $[G:H]$ is a prime p. Prove that $H$ is normal.

Past year exam question: Let $G$ be a group and $H$ a subgroup of $G$. Suppose there exist some $g \in G$, $g \notin H$, such that $gH=Hg$, and $[G:H]$ is a prime p. Prove that $H$ is normal. Sketch:...
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exponent of a finite group divides order of the group

Let $m\in\Bbb N$ be the exponent of a finite group $G\ $ ($|G|=n$). It' the smallest integer such that $g^m=e_G\ \forall g\in G$ Proving that $m\mid |G|$ is the same as proving that the least common ...
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3answers
62 views

How to prove that an even permutation of $A_n$ is a square of another permutation from $S_n$?

I am trying to go through a proof which contains a statement that an even permutation from $A_n$ is a square of another permutation from $S_n$. My basic ideas are like this: Suppose an even ...
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0answers
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Regarding the dimension of irreducible (finite-dimensional) group representations

Ok, I admit it. I'm confused. I'm a physics student attempting to learn some group theory and topology in my spare time. I was reading about group representations. For example I get that the set of ...
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1answer
30 views

Complement of a normal subgroup is single conjugacy class

The dihedral group of order $2n$ with $n$ odd has property that the complement of cyclic subgroup of order $n$ is a single conjugacy class. Q. Are there any other solvable groups in which ...
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1answer
66 views

Show that if $|I| > \frac{3}{4}|G|$ then $G$ is abelian. [duplicate]

I've self studying Nathan Jacobson's Basic Algebra and I came across this question: Let $G$ be a finite group with $\alpha$ a automorphism of the group. Denote, $$I =\{g \in G : \alpha(g) = g^{-1}\}$$...
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31 views

Let $p$ be a prime dividing $|G|$, and let $S_p$ be a $p$-Sylow subgroup of $G$. Show that $N(N(S_p))=N(S_p)$.

Attempt: Trivially $N(S_p) \subseteq N(N(S_p))$ because a subgroup is a subset of its normalizer. Then, let $g\in N(N(S_p))$. Then we have $gN(S_p)g^{-1}=N(S_p)$ by the definition of normalizer. But $...