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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6answers
78 views

How can I formally prove that $3\mid 2^n+1\iff n=2m+1,m \in \mathbb N$

How can I formally prove that two to the power of some n all plus one is divisible by three when n is odd (1,3,5,7,...)? Or, another words,$$2^n+1=3k \Leftrightarrow (n=2m+1,m \in \mathbb N)\bigwedge(...
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1answer
133 views

Solve $x^7=e$ in a group

Let $(G,\cdot)$ be a group having the propery that $\exists a \in G$ such that $ax=x^4a$,$\forall x \in G$. Solve the equation $x^7=e$. I started by observing that for $x=a$ we have that $a^2=a^5$, so ...
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1answer
39 views

How many legal positions are there on a Rubik's cube with two pieces glued together?

Legal here means reachable from the solved state without separating the two glued pieces. Answer depends on the type of pieces glued together (egde+corner or edge+center). Is it just all normally ...
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0answers
46 views

What is a function group (in older literature)?

I am reading Goldberg's 'Invariant transformations, Conservation laws and energy-momentum' from 1980. He frequently uses the notion of a function group. For example: "For any field theory whose ...
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2answers
43 views

elements of order two in $D_{10}$

Which elements have order two in $D_{10}$? In $D_{10}$ there are $10$ elements, five of which are rotations and five reflections. Let $\rho = (1\hspace{1mm}2 \ldots 5)$ and $\tau = (1)(2\hspace{1mm}5)...
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1answer
48 views

Order of a finite multiplicative subgroup of a field

Let $G$ be a finite subgroup of the invertible elements of a field $F$. Show that if char$F\neq0$, then $G$ is cyclic of order $n$ with $n$ prime to char$F$. I have solved the first part but have no ...
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1answer
111 views

Prove that either $A=\Delta B$ or $A=\Delta C$

Suppose that $(A,+)$ is an abelian group and that $A=B \cup C$. Define for any $X \subseteq A$ the following $$\Delta X = \{ x_1-x_2 ; x_1 , x_2 \in X \}$$ Show that, if $B$ and $C$ have non-empty ...
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1answer
41 views

How to show $\frac{\langle x,y\rangle }{H}\cong\mathbb{Z}\oplus \frac{\mathbb{Z}}{2}$?

Suppose an abelian group $K$ is generated by two elements $\langle x,y\rangle$ and $H$ is a subgroup. Now suppose $nx + my \in H \iff n=0$ and $m$ is even. Then $\frac{\langle x,y\rangle}{H}\cong\...
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1answer
45 views

Number of group homomorphisms from a cyclic group to a finite group

$\textbf{Proposition:}$ Let $G = \langle x_0\rangle$ be a cyclic group of order $n\in \mathbb{N}_0$ and $H$ a finite group. Then the number of group homomorphisms $\varphi:G\to H$ is equal to the ...
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2answers
78 views

Irreducible representation of $\mathfrak{so}(4)$

I am supposed to give a 9-dimensional irreducible representation of $\mathfrak{so}(4)$. I know that $\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ and hence I have a 6-dimensional ...
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0answers
31 views

Isomorphism preserves order of elements between two groups. Is the converse true? [duplicate]

Suppose $G$ and $G'$ are two groups such that for all $x$ in $G$ we have a element $x'$ in $G'$ such that $o(x)=o(x')$. Now my question is that whether two groups are necessarily isomorphic? If not ...
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0answers
28 views

Condition on orders of cyclic group so that $x^i \mapsto y^i$ is an homomorphism

So, I'm working my way through Artin, with the help of the wonderful lectures of Professor Gross, and following the required homework he asks in his class, and I'm not completely sure about one of ...
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1answer
67 views

Cyclic group of order 24 [closed]

I have been asked to prove or disprove the following: If $G$ is a group of order $24$ such that $a\in G$ , $a^{12}\ne e$ and $a^{8}\ne e$ where $e$ is the identity of $G$, then the group is cyclic....
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2answers
104 views

Are all nilpotent groups hamiltonian?

Are all nilpotent groups hamiltonian? That is, is every subgroup of a nilpotent group normal? I don't think so. Every Sylow subgroup of nilpotent group is normal and every nilpotent group is a direct ...
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3answers
49 views

Are the canonical injections into the coproduct a structure or a property?

Can someone illuminate why the choice of canonical injections seems to be a part of the structure of a coproduct? Given a category $\mathcal{C}$ and objects $A, B$. Then their coproduct, if it exists, ...
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1answer
21 views

Number of unit roots in a group verifying a special property

Let G be a group verifying that all subgroups are generated by kth powers of G elements for some k. My question : Is there a finite number of pth roots for the neutral ? I tried to study the fibers ...
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1answer
48 views

Meaning of the Sylow's second theorem and related question.

This is from Wikipedia Sylow's Theorem 2 Given a finite group $G$ and a prime number $p$, all Sylow p-subgroups of $G$ are conjugate to each other, i.e. if $H$ and $K$ are Sylow p-subgroups of G, ...
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0answers
90 views

Generalization of normal subgroup

I am wondering whether the following concept appears in the group theory literature under some (perhaps different) name. Let $G$ be a group and let $A,B$ be subgroups of $G$. Definition. Say that $(...
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1answer
51 views

Subcategories of a group

Find all subcategories of a group. Which of them are full? A group is a category with one object in which every arrow is an isomorphism. To specify a subcategory, I need to specify a bunch of objects ...
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2answers
41 views

Normal closure of powerfully embedded subgroups.

I'm reading a paper of Lubotzky and Mann (J. Algebra 105, 1987, 484-505), and Im doubtful in a proof. Proposition. Let $N$ powerfully embedded subgroup of $G$. If $N$ is the normal closure of some ...
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0answers
35 views

In proving Splitting Lemma, why does Aluffi show $N \cong M \oplus \ker \psi$ instead of $N \cong M \oplus \mathrm{coker}\varphi$ like Hatcher?

From Algebra: Chapter $0$ by Aluffi: He proves this by showing $N \cong M \oplus \ker \psi$: However, as I read Hatcher's Algebraic Topology this past semester, he has us show (in this case) ...
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1answer
51 views

Perfect finite groups with balanced presentations

A balanced presentation for a group is a presentation with an equal number of generators and relations. A perfect group is a group that is equal to its commutator group. I am wondering what finite ...
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1answer
29 views

Trouble computing an index

Let $G$ be a group, $N$ a normal subgroup of $G$ such that $G/N$ is infinite cyclic, $C$ an infinite-cyclic subgroup such that $C\cap N=1$, and $\Lambda$ a subgroup of index $2$ of $G$. Let $N_{2}:=...
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0answers
28 views

Is $\langle\mathbb R^*,*\rangle$ a group ($\mathbb R^* =\mathbb R-\{0\}$), where $*$ is defined as $a*b = |a| b$?

Let $\mathbb R^*$ be the set of all real numbers except $0$. Define $*$ on $\mathbb R^*$ by $a*b= | a | b$. $*$ is associative on $\mathbb R^*$ and $1$, $-1$ are left identities and $1$ is right ...
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1answer
55 views

Is this formula for induction always possible?

I am beginning to learn representation and character theory. I worked out some proofs of the character table of the dihedral groups. But I have some issues understanding how deep is the method. Say $n$...
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1answer
81 views

For subgroups $H,K$ of $G$ if $HK$ is always a subgroup, is $G$ abelian?

This is a well known theorem from elementary group theory. Let $G$ be a group and $H,K\leq G$, Then $HK$ is a subgroup of $G$ if and only if $HK=KH$. As a consequence of this, if $G$ is abelian, ...
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0answers
30 views

homeomorphism of quotients involving upper half plane

Let $G = \operatorname{GL}(\mathbb{R}) \times \operatorname{GL}(\mathbb{Q}_p) , K = \operatorname{O}(2) \times \operatorname{GL}(\mathbb{Z}_p)$, $$C=\mathcal{Z}(\operatorname{GL}(\mathbb{R}))\times \{\...
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2answers
82 views

Conjugacy classes of a group of order $8k$

Let G be a group of order $8k$, show that there are at least 5 different conjugacy classes. Hi everyone, I have this problem I think I had a solution involving stabilizers, however I feel there must ...
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1answer
33 views

$H\lhd G$, $|H|=p^k$ then $H$ is included in every $p$-Sylow subgroup of $G$

Let $G$ be a finite group and let $H$ be a normal subgroup of G, with $|H|=p^k$. Then you have to prove that $H \subset P$, for every $P\in\mathrm{Syl}_p(G)$. What I thought is that, as $p$-Sylow are ...
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1answer
31 views

If $(H,+)$ is a subgroup of $(\mathbb{R},+)$ with finite $H \cap [-1,1]$ with some non-zero element, then $H$ is cyclic.

$\mathbf{Question:}$ $(H,+)$ is a subgroup of $(\mathbb{R},+)$ such that $H \cap [-1,1]$ is finite and contains elements other than $0$. Show that $(H,+)$ must be cyclic. $\mathbf{Attempt:}$ Since $\...
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1answer
39 views

How do I show that $B \to A \oplus C$ where $b \mapsto (p(b), j(b))$ is surjective?

From Hatcher's Algebraic Topology: How do I show that $B \to A \oplus C$ where $b \mapsto (p(b), j(b))$ is surjective? I can show that this map is well-defined and we use the fact that $\ker ...
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5answers
60 views

Are the only single element normal subgroups precisely the elements that make up the center of a group?

So, a single element normal subgroup $n$ of group $G$ would be defined as $ \forall g \in G: gng^{-1} = n$. All elements of an abelian group would qualify, and so would more generally the elements of ...
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2answers
261 views

Transitive action of a discrete group on a compact space

Let $G$ be a discrete countable group acting on a compact, Hausdorff space $X$. Assume that the action is transitive. Namely, $G\cdot x=X$, for all $x\in X$. Does it follow that $X$ is finite? I ...
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1answer
33 views

On the Sylow subgroups of a subgroup

What I am trying to prove: Let $G$ be a finite group and $H$ a subgroup of $G$. Then prove that two $p$-sylow of $H$ are contained in different $p$-sylow of $G$. My attempt was suppose that they are ...
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1answer
28 views

Reference request: structure of stabilisers of totally isotropic subspaces in orthogonal (and unitary) groups

I am looking for a book or paper which covers the structure of stabilisers in $GO(n,F)$, $SO(n,F)$ (or maybe in $\Omega(n,F)$) of totally isotropic subspaces of dimension $k$. Can you please suggest ...
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1answer
37 views

Non-abelian finite $p$-group in which every element has prime order? [duplicate]

Let $G$ be a finite group such that for some prime $p$, every element has order $p$ . Can $G$ be non-abelian ? My thoughts: Of-course $G$ is a $p$-group so has non-trivial center. If $G$ is non-...
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1answer
54 views

Sylow subgroup of a subgroup 5

I am aiming for looking for all the Sylow subgroups of classical groups, which gives me a seemingly elementary question: what can we say about Sylow subgroups of a subgroup if we know all Sylow ...
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0answers
26 views

Construct a 2 generator group $G$ that is an extension of a free abelian group $N$ of infinite rank by an infinite cyclic group

Construct a 2 generator group $G$ that is an extension of a free abelian group $N$ of infinite rank by an infinite cyclic group I'm looking for a quick example of this, I've been trying to figure ...
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1answer
40 views

How to show $\mathrm{Hom}_\mathsf{Ab}(\mathbb Z/a\mathbb Z,\mathbb Z/b\mathbb Z) \cong \mathbb Z/(a,b)\mathbb Z$?

With some help, I figured out how to show $\mathrm{Hom}_{R-\mathsf{Mod}}(R/I, N) \cong \{n \in N \mid \forall a \in I, an=0\}$. I would like to use this result for the current problem. We can ...
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0answers
40 views

What are some good books on group theory? [duplicate]

I want to self-study group theory. Does anyone have good recommendations for me?
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1answer
48 views

If $A\oplus B\approx A$ then $B=0$? [duplicate]

Suppose $A,B$ abelian groups, such that $A\oplus B\approx A$, can I conclude that $B=0$? If it's true, is there any hint how to prove it?
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17 views

Walk generating function for adjacency matrix

In its book "Random Walks on Infinite Graphs and Groups", W. Woess defines the Green function (or walk generating function) as $$G(x,y|z) = \sum_{n = 0}^{\infty} p^{(n)}(x,y)z^n, \quad x, y \in X, z \...
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2answers
43 views

Injective map of abelian group and product of cyclic quotients

Let $A$ be an abelian group. We have a map from $A \to \prod{(A/I)}$ where $A/I$ varies over the cyclic quotients of $A$. This map is given by sending $x$ to $\prod (x \text{ mod } I)$. Is this map ...
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0answers
36 views

Cocycles and group extensions

I'm trying to understand how elements in the second cohomology group with coefficients in some other group correspond to group extensions. This is what I understand: Suppose we have two (countable) ...
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1answer
62 views

Why does $C_G(g)$ have index $q$ if $|G| = pq$ and $|g| = p$ for all nonidentity $g \in G$?

I'm studying Abstract Algebra, 3rd Edition by Dummit and Foote. In Section 4.4 (Automorphism), one of the examples proves that $G$ is abelian if both of the following hold. $|G| = pq$ where $p$ and $...
5
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2answers
125 views

A finite union of infinite cyclic subgroups of a group $G$ is never a group.

$\mathbf{Question}$: Consider an infinite group $G$. Let $\langle a_1\rangle , \langle a_2\rangle, \ldots, \langle a_m\rangle $ ($\langle a_i\rangle =C_i$) be a finite collection of infinite cyclic ...
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1answer
31 views

If $r = \frac{1}{|G|} \sum_{g \in G} \chi(g^{-1})g$, then $r^2 = 1/(\chi(1)) r$ in the group algebra

Suppose we have a finite group $G$ and an irreducible character $\chi$ of $G$. Now, define in $\mathbb{C} G$ (the group algebra / group ring of $G$), the element $$r = \frac{1}{|G|} \sum_{g \in G} \...
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0answers
18 views

Help with disjoint cycle decomposition and finding image/pre-image of a S8 group

i have this exercise and i'm confused about the part where i find the image and pre-image. I solved it for 5 in alfpha, found the image and preimage (preimage is number that 5 results from and image ...
1
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1answer
50 views

Finding a Group isomorphism G

According to wiki, Isomorphism: Given two groups $(G, ∗)$ and $(H, {\displaystyle \odot })$, a group isomorphism from $(G, ∗)$ to $(H, {\displaystyle \odot })$ is a bijective group homomorphism ...
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1answer
29 views

Cycle type of a permutation in $S_n$ and its relation to partition of $n$ and its Young diagram

I know that it's possible to assign to each permutation its cycle type. I found two definitions of the cycle type and its relation to a partition of $n$: First definition Given $\sigma \in S_n$ ...