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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How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : \begin{bmatrix}4\mathbb{Z}+1&8\mathbb{Z}\\4\mathbb{Z}&4\mathbb{Z}+1\end{...
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How to find out generators of the following free group?

The following is the quotient group of SL($2,\mathbb{Z}$). Consider $(H/\{-1,1\} \cap H)$ where $H=\begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}$ How do ...
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how to find the index of following subgroup?

if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$. or I= \begin{bmatrix}2\mathbb{Z}+1&2\mathbb{Z}\\2\mathbb{Z}&2\...
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An exercise on isomorphic groups of order 24

I am a beginning learner of group theory. Which of the following are isomorphic:$$\mathbb{Z_{24}}, D_{4}\times \mathbb{Z_{3}},A_{4}\times \mathbb{Z_{2}},\mathbb{Z_{2}}\times D_{6}, \mathbb{Z_{12}}\...
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Morphisms of a category with one object, which is a group

I've read two articles(1,2), but still have a question about structure of group, say $G$, as an one-object category. Let's call this category $C$. I understand that morphisms of $G$, which is ...
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42 views

Prove that $g\in P$

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup. If $g$ is an element of $G$ with order a power of $2$ then it lies in some conjugate of $P$. I get this. But if also $g\in C_G(P)$, then it is ...
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44 views

Expressing $z\in G$ as $z=gh^2$ where $g$ is a $2$-element and $|h|$ is odd [closed]

Let $z\in G$ where $G$ is a finite group, then is it always true that there exist elements $g,h\in G$ such that $z=gh^2$ where $|g|=2^k$ for $k \in \Bbb{Z_{\ge0}}$ and $|h|$ is odd?
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40 views

Showing a subgroup contains the identity element

Let $G$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ along with $+$. Show that $H$ defined by $H=\{f:f(x)=0 \text{ for all } x \in [0,1]\}$ is a subgroup. I am able to show $H$ has ...
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23 views

All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
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1answer
48 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\theta \colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi ...
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Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group $S_3$...
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22 views

Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
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64 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides $$\gcd(\...
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Multiplying elements of a multiplicative cyclic group

This is about correct operation in a multiplicative cyclic group. Say we have the group $\mathbb Z_4^{\times} = \{1,2,3\}$, if we multiply the elements, say $3\cdot 2\cdot 2 $ we get $12 \mod 4 = 0$ ...
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41 views

Characteristic subgroups of order $2$

Could anybody give an example of a finite abelian $2$-group with more than one characteristic subgroup of order $2$ ? (In other words, a finite abelian $2$-group $G$ with a Klein subgroup $V$ such ...
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Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
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1answer
22 views

normaliser of dihedral group of order 4

Let $D_{2n}$ be the dihedral group of order $2n$. I would like to show that if $4$ divides $n$, then the normaliser $N_{D_{2n}}(D_{4})=D_{8}$. I know that $$|D_{2n}:N_{D_{2n}}(D_{4})|=|\mathcal{C}|,$$...
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$G$ is a finite group, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$.

Prove / disprove: Let $G$ be a finite group of order $n$, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$. I can't find a group that will satisfy this condition so I think it's ...
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Stabiliser of a non-isotropic 1-space in $\Omega(n,q)$

Let $n,q$ be odd, $V$ be the $n$-dimensional vector space over $\mathbb{F}_{q}$, and consider the subgroup $$G=\Omega(n,q)=\{r_{v_{1}}r_{v_{2}}\dots r_{v_{k}} : k \textrm{ even }, \prod_{i=1}^{k}{(v_{...
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reference request Schur Zassenhaus Theorem

I am looking for a reference for the Schur Zassenhaus Theorem, saying that any normal Hall subgroup admits a complement. An on-line search show that it is supposed to be in "The theory of groups" by ...
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36 views

Is this a linear transformation? in the context of group representations

Let $G$ be a group. A regular representation is given as $V=\mathbb{C}[G]$, a vector space, where $l: G \to GL(V)$ be the action is given by $l(g)(\alpha)(h) = \alpha (g^{-1}h)$ for all $g,h\in G, \...
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42 views

Subgroups of $S_{10}$ that contain $\langle \sigma = (123)(456)\rangle$

In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$. I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle ...
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Proving that a homomorphism between two rings is surjective

The problem: ($\mathscr{F}(\mathbb{R})$ is the set of real valued functions) Let $\phi:\mathscr{F}(\mathbb{R})\to\mathbb{R}\times\mathbb{R}$ be a function defined by $\phi(f)=(f(0),f(1))$ Prove ...
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1answer
22 views

Semidirect product when the action factors

Suppose I am forming a semidirect product of finite groups $G \rtimes_\theta A$. Here, $\theta : A \to \mathrm{Aut}(G)$ is a homomorphism. Now, suppose I notice that the homomorphism $\theta$ factors ...
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159 views

Proof that the Rubik’s Cube group is 2-generated

Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube: Frank Barnes observes that the group of the cube is generated by two moves: \begin{align*} \alpha &= L^2 B R D^{-1} L^...
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Regarding part of proof of proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space.

Proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space. Part of Proof: Let $x$ and $y$ be distinct points of $G$. Then $x^{-1}y \neq e$ (identity element). ...
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Determine if quotient group $S_4/N$ is isomophic to $S_3$ [duplicate]

Let $N = \{1,(12)(34),(13)(24),(14)(23)\}$. Determine if the quotient group $S_4/N$ is isomophic to $S_3$. I computed the cosets: $N, (12)N, (13)N, (14)N, (123)N, (234)N$, and the others are ...
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65 views

Is the following set a group?

Let $ G= \begin{pmatrix} a & a\\ a & a\\ \end{pmatrix} $ where $a\in \Bbb R, a \neq0$. I need to show that $G$ is a group under matrix multiplication. The ...
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Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
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1answer
138 views

How to count the closed left-hand turn paths of planar bicubic graphs?

When you draw a planar cubic bipartite graph $\Gamma$ and 3-color its edges you can use this as an orientation $\mathcal O$. Definition A left-hand turn path on $(\Gamma, \mathcal O)$ is a closed ...
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Solvability of nilpotent groups

I'm uncertain about my proof about this exercise regarding nilpotent groups. If someone could me help me out, that would be appreciated. There's a post about this problem, but it uses another ...
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Pro-alternating completion of a free group

Let $F$ be a free group with $r$ generators. Then for every $n \in \mathbb{N}$ there are exactly $H_n = \left(\frac{n!}{2}\right)^r - 1$ nontrivial distinct homomorphisms from $F$ to $A_n$, the ...
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31 views

Generating set of a group and even subgroup.

Let $G=\langle S\rangle$. Now let $H=\{x_1x_2\cdots x_m\mid x_i\in S \cup S^{-1},\ i\le m\in\Bbb N,\ m\text{ is even}\}$ I'm trying to prove that $[G:H]=1$ or $2$. I started doing this by proving ...
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Why is $M_{21}\cong PSL_3(\mathbb{F}_4)$?

I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, $...
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Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$ $H$ is the set of fixed points on $A$ $A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, ...
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1answer
22 views

Prove the set of permuations which permute only finitely many elements is a normal subgroup

Let $A$ be a non-empty set and let $X$ be a subset of $S_A$ Now let $F(X) = \{a \in A : \sigma(a) = a, \forall \sigma \in X\}$, $M(X) = A\backslash F(X)$, and $D = \{ \sigma \in S_A : \mid M(\sigma)\...
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In a Group, is the existence of the left identity equivalent to the existence of the unique two sided identity?

I've read many definitions in different books, and some of them specifically point out in their definition the existence of left inverse, left identity, and associativity. But the grand majority does ...
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$H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\} , \forall g \in G \setminus H$ , then $|\cup gHg^{-1}|>\dfrac 12 |G|+1$

Let $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\}$ for all $g \in G \setminus H$. Then is it true that $$|\cup_{g \in G \setminus H}gHg^{-1}|>\dfrac 12 |G|+1$$ If ...
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Finite Groups With Exactly Two Automorphisms [duplicate]

Is there an easy way how to characterize the groups with exactly two automorphisms? I was able to find the following finite groups with exactly two automorphisms: $\mathbb{Z}_3$, $\mathbb{Z}_4$, $\...
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1answer
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Groups With Exactly One Maximal Subgroup

I understand that when $G$ has exactly one maximal subgroup (inclusion-wise), then $G$ has to be cyclic. But is it possible to determine all possible groups with exactly one maximal subgroup?
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1answer
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How can I check whether the group $[16,13]$ in GAP with $3$ generators can be generated by $2$ elements?

The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same ...
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On groups with presentations $ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $…

$$ \langle a,b,c|a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group. What about the following ...
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66 views

Which non-abelian finite groups have the property that every subgroup is normal?

If $G$ is an abelian group, every subgroup $H$ of $G$ is normal. I searched for non-abelian finite groups $G$ , such that every subgroup is normal and GAP showed only the groups $G'\times Q_8$ , ...
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27 views

G-set of 8 colors and 6 sides

"How many distinguishable wooden cubes can be painted if u use 8 colors (different colors on every side)" I have solved this question using Burnside's lemma https://sv.wikipedia.org/wiki/...
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1answer
65 views

Prove that (01) and (01…n-1) generate Sn? [duplicate]

Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1). I understand that this can be solved using induction, and I've set up my base cases. However,...
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Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} $...
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Only two groups of order $10$: $C_{10}$ and $D_{10}$

Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
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124 views

“conjugate to/with” or “conjugated to/with”, a terminology question in group theory.

This is a terminology question from a non-native English speaker. Let $G$ be a group and $a,b\in G$ such that there exists $c\in G$ verifying : $$b=cac^{-1} $$ I could say : the element $a$ is ...
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25 views

Inverse element in semidirect product

If K and Q are both groups and $h:Q\rightarrow \text{Aut}(K) $ is a homomorphism then the group operation for the semidirect product $K\rtimes_hQ$ is: $$(k_1,q_1)*(k_2,q_2)=(k_1h(q_1)(k_2),q_1q_2)$$ ...
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28 views

How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$?

This was a question in our quiz today and no one in class knew how to answer it correctly or are not sure). How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$, where $U_8$ is ...