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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Defining a group of symmetries with restrictions

For the following group of symmetries, f is not allowed; $$D=\{e,m,m^2,fm,mf^2m | m^2=mf^2m=e\} f \notin D$$ How do we define it or is it even possible to define as a group?
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Subgroup of $S_6$ which is isomorphic to $S_3$

I have to construct a transitive subgroup of $S_6$ which is isomorphic to $S_3$. I was thinking of using Cayley's theorem, but I have no clue how to construct such a transitive subgroup. I know the ...
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center of a finite index subgroup of a mapping class group is trivial?

(Sorry for the long read, you may skip to the very end, to see the question there.) I am reading an article and trying to understand the proof, but cannot work out a crucial detail. Here's the ...
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Show that $G / H \simeq G^{\prime} .$ Hence show that there does not exist any homomorphism from $Z_{9}$ onto $Z_{6}$

Question: Let $G$ and $G^{\prime}$ be two groups and $\phi: G \rightarrow G^{\prime}$ be an onto homomorphism. Let $H=\ker\phi$. Show that $G / H \simeq G^{\prime}.$ Hence show that there does not ...
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When are free abelian normal subgroups “virtually” central?

See my previous question for the motivation for this question: Are infinite cyclic normal subgroups "virtually" central? Let $G$ be a (finitely generated) group and let $K \lhd G$ be a ...
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Doubt in dummit foote - direct subgroup.

In Section $5.2$ of dummit and Foote, the fundamental theorem of finitely generated ableian group is stated as A finitely generated abelian group $G$ is isomorphic to $\mathbb{Z}^r \times \mathbb{Z}_{...
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Determining the order of matrices in $SL(2, \Bbb R)$

Problem Determine the order of the matrices: $$ A = \begin{bmatrix} \cos(60°) & -\sin(60°) \\ \sin(60°) & \cos(60°) \\ \end{bmatrix} \space B= \begin{bmatrix} \cos(\sqrt 2°) &...
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Why are these two exact sequences “essentially the same”? (Lang's Algebra, pp. 15--16)

A question here concerns much the same issue, but I don't understand the explanation there, so I will ask here for a more detailed explanation. Excerpt: Discussion: I think I understand the ...
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Are infinite cyclic normal subgroups “virtually” central?

Let $G$ be a finitely generated group and let $C$ be an infinite cyclic normal subgroup of $G$. My question is: Does $G$ necessarily have a finite-index subgroup $H$ such that $C \le Z(H)$? If not, ...
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Prove that there is a binary vector that remains constant under multiplication from a group of binary matrices

Suppose $G$ is a group of binary 7$\times$7 matrices and $|G|=64$. We want to prove that there exists a non-zero binary vector of length 7, $v$, such that for all element $g \in G, gv = v$. I have no ...
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Subgroups of order $p^2$ of $Z_{p^2} \times Z_p$

How many subgroups of order $p^2$ does the group $Z_{p^2} \times Z_p$ have? Here $p$ is a prime and $Z_{k}$ is the cyclic group of order $k$ (NOT the $\mathbb{Z}_k = \mathbb{Z} / k\mathbb{Z}$). One ...
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Closure of $\langle\sigma\rangle$ in $\text{Gal}(\bar{\mathbb F}_q / \mathbb F_q)$

We know that $$G_n = \text{Gal}({\mathbb{F}}_{q^n} / \mathbb{F}_q) \cong \mathbb Z / n \mathbb Z,$$ where the Frobenius $\sigma_n$ of $\mathbb F_{q^n}$ is mapped to $1 \in \mathbb Z / n \mathbb Z$. $G ...
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2answers
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Difficulty in proving that S_3 is isomorphic to the free group on two letters with the following relation:

Using the universal property of the free group, I want to show that $$S_3 \cong G = \langle a,b: a^3=b^2=e;ba=a^2b \rangle.$$ I think I understand the general idea of how to show that a group is ...
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conjugacy in symmetric group

I was reading Dummit and Foote and encountered the following statement: any two elements in $S_n$ are conjugate if and only if they have the same cycle types. However, I am able to produce a counter ...
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normal subgroups of Alternating group

Klein's $4$ group, $\{1\}$ and $A_4$ are three normal subgroups of $A_4$. I want to know about any other normal subgroups of $A_4$. Do these exist?
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Group acting coprimely by automorphism

Consider that a group $A$ acts by automorphism on a finite group $G$. If this action is coprime, i.e. $\gcd(|A|,|G|)=1,$ can we affirm that this action is fixed point free, i.e. $C_G(A)=1$? I tried to ...
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1answer
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solvable group and abelian

From wiki, a group $G$ is solvable if there are subgroups $1 = G_0 \trianglelefteq G_1$⋅⋅⋅$\trianglelefteq G_k = G$ such that $G_{j−1}$ is normal in $G_j$, and $G_j /G_{j−1}$ is an abelian group, for $...
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Show that $(S,*)$ is a group,where $S=\left\{a,b,c,e\right\}$

Given a set $S=\left\{a,b,c,e\right\}$ equipped with a binary operation $*:S \times S \to S$,such that $$a^2=b^2=c^2=abc=e.$$ Show that based on the table $$ \begin{array}{r|rr} *&e&a&b&...
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1answer
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Use Burnside's theorem to find the colourings of an octahedron

The question is to find in how many ways we can colour the edges of an octahedron with $k$ colours by using Burnside's theorem. I already know that I'm supposed to find the automorphism group to get $|...
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Universal property quotient

Let $G$ be a group and $N$ a normal subgroup of $G$. A pair $(K, \pi: G \to K)$ where $\pi$ is a group morphism is said to satisfy the universal property of the quotient group if for all group ...
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Prove that $G$ is commutative knowing that $G$ is a group.

I have the following question I have to prove/disprove: Let $(G, *)$ be a group. If for every $a, b ∈ G$ we have $(a * b)^6 = a^6 * b^6$ then $G$ is commutative. I tried: I know that we have this ...
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Classification of regular connected covering spaces.

I read the following: By classical covering space theory, the connected, regular covers of a $CW-\text{complex}$ are classified by the quotients of its fundamental group. Aren't connected covering ...
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Let $p$ a prime number and $k\in \mathbb{Z}_+$. If $|G|=p^k$ and $H<G$ ; $|H|=p^{k-1}$, then H is normal in G. [closed]

I just started studying abstract-algebra some weeks ago and I was doing an exercise list that was asking if it is true or false. I mean, I didn't find any counterexemple but I couldn't prove either, ...
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Sufficient conditions for a group to be a free group

Let $U$ be a group with generators $\{a_1,\dots ,a_r\}$. If we want to show that $U$ is a free group, is it sufficent to show that $a_{i_1}^{k_1}a_{i_2}^{k_2}\dots a_{i_l}^{k_l} \neq 1$ where $k_i \in ...
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Why every element of $G$ lies in some coset $\tau_i H$?

I'm reading Lang's "Undergraduate Algebra". Specifically here: I understand why $\sigma$ lies in $\tau_i H$ but it's not clear to me why we have shown every element of $G$ lies in some ...
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Let $G$ be a finite group, $S$ a sylow subgroup of $G$ and $H<G$ such that $N_G(S)\subseteq H$.

Let $G$ be a finite group, $S$ a sylow subgroup of $G$ and $H<G$ such that $N_G(S)\subseteq H$. (1) Show that for all $x\in N_G(H), xSx^{-1}$ is a sylow subgroup of $H$. (2) Show that $N_G(H)=H$. ...
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Show that the unit quaternions form a group.

Here is the question I want to answer $(d)$ in it: Define $E \in GL_{2}(\mathbb{R})$ by $E = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and let $\mathcal{R} = \{aI + bE| a,b \in \mathbb{R}...
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Are upper triangular matrices with real entries associative?

I'm trying to disprove that with a,b,c belonging to the real numbers, that the matrix \begin{bmatrix} a & b \\ 0& c\end{bmatrix} is NOT a group under matrix multiplication. With the criteria ...
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Generation of $SU(n,q^2)$ by subgroups isomorphic to $SU(2,q^2)$

Throughout, let $q$ be an odd prime power. Let $GF(q^2)$ be the field with $q^2$ elements. My question concerns the generation of the special unitary group $SU(n,q^2)$, where $n \ge 2$, by certain ...
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1answer
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normality in conjugate subgroups [closed]

Let $G$ be a finite group, let $K$ and $gKg^{-1}$ be conjugate subgroups of $G$ (for some $g \in G$), and suppose that some subgroup $H \subseteq G$ is contained in both $K$ and $gKg^{-1}$. If $H$ is ...
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1answer
33 views

Finitely presented group with intermediate Turing degree word problem

Does there exist a finitely presented group with undecidable word problem, but so that an oracle to solve the word problem for this group wouldn't be sufficient to solve the halting problem in general?...
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1answer
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Isomorphisms of $S_n$ [closed]

I am to find the number of isomorphisms between $S_n \to S_n$. How do I proceed? One is the map which maps all the elements to itself. So if $\varphi(ab)\to (cd)$ will be one case. Consider the case ...
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4answers
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If $H \leq G$ and $\ker (f) \subseteq H$, then $f^{-1} (f(H)) = H$

If $H \leq G$ and $\ker (f) \subseteq H$, then $f^{-1} (f(H)) = H$. This is as far as I got: $H \leq G \implies \forall a,b \in H, ab^{-1} \in H.$ ker($f$) $\subseteq H$ equivalently states $f(x) = e \...
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Why is so obvious that an element of a group is the product of finitely many generators? [closed]

A colleague said to me that it is by definition of a group. Thinking about that is giving it more sense; but, does anyone have a clarifying explanation? Thanks in advance :)
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Let, $N,K,N',K'$ be groups and $N\cong N',K\cong K'$. Does $N\rtimes K\cong N'\rtimes K'$?

I have tried this problem in the following way- Let $f:N\to N'$ and $g:K\to K'$ be two isomorphisms. $K, K'$ act on $N,N'$ respectively such that $k•(n_1n_2)=(k•n_1)(k•n_2)\ \forall k\in K, n_1,n_2 \...
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1answer
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For a group homomorphism $f: G \to G'$, show that if $H < G$, then $f(H) \leq G'$.

This is an exercise from "Introduction to Abstract Algebra" by Timothy J. Ford. I really don't know where to start proving this, so if you could give me a start on how to do this, but not ...
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Is $A_{33}$ a unique subgroup of $A_{34}$ with index $34?$

As $A_{33}$ is a unique subgroup of $S_{33}$ with index $2$ and $A_{33}$ is a normal subgroup of $S_{33}$, then $A_{33}~{\rm char}~S_{33}$? As $S_{33}$ is a unique subgroup of $S_{34}$ with index 34, ...
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finding quotient group relationship [duplicate]

The subgroup $S_1$ (the unit circle) in the multiplicative group of complex numbers without 0 is a normal subgroup. Identify the quotient group $\mathbb{C}^\times / S_1$. I thought the quotient group ...
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1answer
30 views

coset representative proof

I'm studying coset representatives and I don't understand part of a proof. To be specific, it's about proving "Any $x \in gH$ is a coset representative" The proof is $x \in gH \to g^{-1}x \...
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why showing that $\mathcal{R} \cong \mathbb{C}$ as rings implies that $\mathcal{R}$ is a field?

Here is the question: Define $E \in GL_{2}(\mathbb{R})$ by $E = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and let $\mathcal{R} = \{aI + bE| a,b \in \mathbb{R}\} \subset M_{2}(\mathbb{R})....
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1answer
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Quick Question on Quotient Space

Let $X$ be a normed linear space and let $U \subseteq X$. What is the quotient space $X/ U $like when $X=U$? Is it just like $X$? So is it trivially a normed linear space? Thanks!
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Explicit automorphism map of $SO(n;\mathbb{R})$ for $n \neq 2,8$?

How do we construct a precise map of inner + outer automorphism of special orthogonal group $SO(n;\mathbb{R})$? $d=2$; We can look at $SO(2;\mathbb{R})=U(1)$ which is abelian, and we know the inner $$...
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1answer
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“Short exact sequence split” answer illustration.

Here is the question from this link Short exact sequence split For groups $G$, $H$, and $K$, assume there exists a left-split short exact sequence: $$ 1 \rightarrow K \xrightarrow{\varphi} G \...
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Showing $g(A \cap B)g^{-1} = g(A)g^{-1} \cap g(B)g^{-1}$ in a group.

Let $G$ be a group, $g \in G$, and $A \cap B$ are subgroups. Is the following true? $$g(A \cap B)g^{-1} = g(A)g^{-1} \cap g(B)g^{-1}$$ I managed to prove that $g(A \cap B)g^{-1} \subseteq g(A)g^{-1} \...
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$O(d,1)$ isomorphisms to $Sp(n;\mathbb{F})$ for some fields $\mathbb{F}$ and $n$

The following orthogonal groups claimed to have the isomorphism (see the Abstract of this paper or here PDF) to Symplectic group: $$O(2,1) \simeq Sp(2;\mathbb{R})?$$ $$O(3,1) \simeq Sp(2;\mathbb{C})?$$...
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2answers
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When is $H^m=\{ h^m \mid h\in H\}$ a subgroup $H$?

Let $H$ be a group and $H^m=\{ h^m \mid h\in H\}$. I know that this is a subgroup of $H$ when $H$ is abelian. But I want to know that what happens if $H$ is not abelian. For which $n$, $H^n$ is a ...
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1answer
49 views

Order of an element in $\mathbb{C}\setminus \{0\}$

I was wondering the order of an element $z\in \mathbb{C}\setminus \{0\}$ For $z=a+bi$, would it be the absolute value/modulus? That is, $$|z|=\sqrt{a^2+b^2}.$$ Thanks
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1answer
108 views

Examples of finite groups $(G,\cdot)$ where multiplication is string concatenation followed by a 'put back to standard form' algorithm.

The title of the question motivates the specific mathematical question given in the next section. Let $A$ be finite set. For an integer integer $n \ge 0$, a function $s: \{k \mid k \le n \land k \gt ...
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30 views

What is the group generated by the exponential of multivectors?

It is known that $$ \exp : \mathbb{M}(n,\mathbb{C}) \to \operatorname{GL}(n,\mathbb{C}) $$ This relationship only works over the complex field. My question is: $$ \exp : \mathcal{G}_n(\mathbb{R})\to ? ...
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1answer
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How can non-split automorphism extensions exist?

Given a finite simple group $S$, we can consider its automorphism group ${\rm Aut}(S)$. Since ${\rm Inn}(S) \lhd {\rm Aut}(S)$, and $S \cong {\rm Inn}(S)$, we can ask whether $S$ has a complement in ${...

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