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.

33,537 questions
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Structure of сonjugacy subgroup intersection graph

Let $G$ be a finite group, $H$ is proper subgroup and ${\cal H}(H)$ the set of all subgroups of conjugate H. Construct the graph $\Gamma$, with vertices ${\cal H}(H)$ and two subgroups adjacent if ...
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I have troubles understanding this lemma I found in a paper on maximality of non nearly normal subgroups. Statement: Let $G$ be a group satisfying $Max-nn^{-}$. Then the commutator subgroup $G'$ of $... 1answer 34 views Lower bounds on the number of subgroups of a specific group I am visiting this thread:$G$is Abelian iff all the Sylow subgroups of$G$are normal I am wondering whether more can be said about the lower bounds of number of subgroups of$G$when$G$is of ... 0answers 38 views Cayleys Theorem worked example Find the smallest n such that$S_4 \times S_8$is isomorphic to a subgroup in$S_n$. Answer By Lagrange, we know that the minimum$n$has to be at least$12$. But also if$n=12$then this is easy to ... 0answers 54 views A special case of group extensions, other then the semidrect product. It's a well-known fact that if$A$an Abelian group and$G$is a group, then all group extension of$G$by$A$is isomorphic with the group ($A\times G,\,\bullet)$, where the group operation$\bullet$... 1answer 20 views A relation between FC groups and locally finite groups It is proved that if$G$satisfies maximality on non-nearly normal subgroups, denoted with$Max-nn^{-}$and$K$is a cyclic subgroup of$G$, then$K^{G}$satisfies the maximal condition on subgroups. ... 1answer 32 views Centralizer of Sylow$p$-subgroups for nonabelian finite simple groups Given a nonabelian finite simple group$G$, is it always true that $$C_G(P)\leq P$$ for each Sylow$p$-subgroup$P\leq G$? 1answer 43 views The Alternating Group$A_n$We know that$\forall n\geq 5,$the alternating group$A_n$is simple. But$n\leq 4,$Is the alternating group$A_n$simple? I can't find examples of them . Please help me! Thank you. 0answers 14 views FC-groups with maximality on non-nearly normal subgroups I am reading this proof about characterization of FC-groups having maximality on non-nearly normal subgroups, which I will denote with Max-nn$^{-}$. Recall that$H$is a nearly normal subgroup of a ... 0answers 35 views What is the name of the group? What is the name of the subgroup of the affine plane group generated by the transformations of the form $$x \mapsto a \begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta &\cos\theta \end{... 1answer 30 views Normal subgroup but not characteristic subgroup I know every characteristic subgroup is normal subgroup but converse is not true. I find an example in Klien 4- group. But I have seen on internet that in Q_8, each of cyclic subgroup of order 4 ... 3answers 34 views Example of proper image of center of group under epimorphism. Let f:G \to H be a group epimorphism and let Z(G) be a center of a group G. We already know that f(Z(G)) \subseteq Z(H). Is there any small example showing f(Z(G)) is a proper subgroup of Z(... 1answer 59 views Let G be a group of order 8 and y be an element of G of order 4. Prove that y^2 \in Z(G) [duplicate] The question Let G be a group of order 8 and x be an element of G of order 4. Prove that x^2 \in Z(G) already posted here. But the answer is not given there. SO I have tried to solve the ... 1answer 64 views Center of a group with order p^aq^b Are there any examples of groups, G, such that |G|=p^aq^b (where p and q are distinct primes and a,b\geq 1) and |\mathrm{Z}(G)|=p^a? (\mathrm{Z}(G) denotes the center of G) I ... 1answer 51 views Universal Property for Presentations Let G be a group with a presentation \langle S | R \rangle ( G = \langle S|R\rangle), with an associated map f\, : S \rightarrow G Let \bar{N}_{R} be the normal closure of R in F_{S}. ... 1answer 52 views Is there a cyclic subgroup of A_8 of order 10 I'm trying to check if there is a cyclic subgroup of A_8 of order 10. I believe that it does not have one. I think that we can prove it by showing that it can't get build. The problem is more ... 1answer 46 views Product, homomorphism and kernel of these two additive matrices groups [closed] Consider the additive groups \mathit{G}:= \mathfrak{M}_{\mathrm{4x1}}(\mathbb{Z_{11}}) and \mathit{H}:= \mathfrak{M}_{\mathrm{3x1}}(\mathbb{Z_{11}}) (column matrices with height 4 and 3 ... 0answers 33 views Are there nontrivial subsets of the complex unit circle satisfying the multiplicative Jacobi identity? Let a^{(b^c)}\times b^{(c^a)}\times c^{(a^b)}=1 Then a set S satisfies the multiplicative Jacobi identity if this is true for all a,b,c\in S. S=\{1\} satisfies the identity. S=\{0\} doesn'... 2answers 38 views Union of two characteristic subgroups I have proved intersection of two characteristic subgroup is characteristic but when I want to prove this for union I got stuck. Is union of two characteristic subgroup is characteristic subgroup? ... 1answer 49 views The fundamental group of the gluing of two genus g 3-dimensional handlebodies I was given the problem above, and I'd appreciate some help. I think I have a general direction, but I'm not entirely sure if what I'm doing is true, so it'd be great if someone could tell me if what ... 3answers 38 views Finding \tau\in S_9 for \tau(1,2)(3,4)\tau^{-1}=(5,6)(1,3) I'm trying to understand where I'm wrong in my solution. I would like to find all \tau \in S_9 so$$\tau(1,2)(3,4)\tau^{-1}=(5,6)(1,3)$$Meaning - (\tau(1),\tau(2))(\tau(3),\tau(4))=(5,6)(1,3). ... 1answer 47 views If G has a unique subgroup H of a given (finite) Index, then Prove that H is characteristic subgroup of G I know if a group G has a unique subgroup H of a given order then H is characteristic subgroup of G as every automorphism f on G order f(H) = order H . But here H is unique subgroup of G of given ... 1answer 61 views Explicit description of the conjugation action of [[1,0],[0,p]] on the amalgam SL(2,\mathbb{Z})*_{\Gamma_0(p)} a_pSL(2,\mathbb{Z})a_p^{-1} Let a_p denote the matrix [[1,0],[0,p]], where p is prime. Then SL_2(\mathbb{Z}[1/p]) can be presented as the amalgamated product$$SL_2(\mathbb{Z})*_{\Gamma_0(p)} a_pSL_2(\mathbb{Z})a_p^{-1}... 1answer 56 views Finding coproducts in$\mathbf{Grp}$using presentations Aluffi II.8.7 suggests proving that, given groups$G, G'$admitting representations$(A \mid R), (A' \mid R')$respectively, where$A, A'$are disjoint, their coproduct in is$(A \cup A' \mid R \cup R'...
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If you do modulo arithmetic on a circle you have a certain group. But if you fourier transform a function on a circle you get discrete values. (Compared with continuous fourier tranform of a function ...
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Klein four groups, cyclic groups

A basic definition question: "the Klein four group V is the simplest group that is not cyclic." Does this simply mean you need two (non-identity) elements of the group to generate the entire group? ...
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Visualization of the relationship between Conjugacy Classes and Centraliser

I've recently come across a theorem in my abstract algebra course which states that for a finite group $G$ and for any $a\in G$ $|\text{cl}(a)|=[G:C(a)]$ where $\text{cl}(a)$ is the conjugacy class ...
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Can all groups be thought of as the symmetries of a geometrical object?

It is often said that we can think of groups as the symmetries of some mathematical object. Usual examples involve geometrical objects, for instance we can think of $\mathbb{S}_3$ as the collection of ...
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Uniqueness of Rank of Free Group via Abstract Nonsense

It is a well known result that given two isomorphic free groups, their freely generating sets have the same cardinality, which is then called the rank of a free group. I am well aware of the usual ...
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Every subgroup of cyclic group is cyclic,How do I visualize this fact graphically? [closed]

I want to visualize everything in Mathematics.Is there a way to visualize the theorem I stated above?Can it be represented graphically?I want to be very analytical in this topic.So please can anyone ...
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On finite simple groups whose orders are perfect powers

A short note in Group Atlas v2.0 states: $\mathrm{PSp}(4,7)$ is the smallest simple group whose order is a proper power. Question: Is there other known finite simple groups whose orders are ...
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how many abelian transitive subgroups of $S_{n}$

It's well-known that any abelian transitive subgroup A of a symmetric group $S_{n}$ has order $n$. Moreover, does anyone know how many abelian transitive subgroups of $S_{n}$ and what do they look ...