# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $g(A \cap B)g^{-1} \subseteq g(A)g^{-1} \... 0answers 29 views ###$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: $$O(2,1) \simeq Sp(2;\mathbb{R})?$$ $$O(3,1) \simeq Sp(2;\mathbb{C})?$$... 2answers 70 views ### 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 ... 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 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 ...
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 ? ...
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 \${...