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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Let N = Fit(G). Why $ N = O_{p}(G) $ and $ A \leq Z(N) $?

Let $ G $ be a soluble group and $ A $ be a minimal normal subgroup of $ G $,where $ A $ is an elementary abelian group of prime power order. Let each chief factor of $ G/A $ has order $ 4 $ or a ...
4
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1answer
36 views

Presentations of the unity group

I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for ...
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1answer
43 views

Isomorphism under integer multiplication [closed]

Find an isomorphism between the groups $$G_1=\{1,4,8,10,14,16\}\pmod{18}\text{ and } G_2=\{3,6,9,12,15,18\}\pmod{21}.$$ I managed to show that these two are groups but cannot find an ...
2
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0answers
36 views

The group $G$ has order 56. Show that it contains normal Sylow p-subgroup.

I'm new at this topic, so I'm not sure about whether is my solution acceptable or not. Could you check it? By Sylow theorem there exist $|P|=7$ and $|Q|=8$. Sylow $p$-subgroup is normal iff it is ...
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4answers
48 views

Prove that the 4-group V is normal subgroup of $S_4$ by using isomorphism theorem

Prove that the 4-group V is normal subgroup of $S_4$ First, by using the multiplication table, I am able to prove that 4-group V is subgroup of $S_4$. But I face problem in proving that $\forall ...
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2answers
45 views

Prove that if $G$ has a subgroup of order $n$, then $H$ has a subgroup of order $n$.

Would someone please help me: Question : Let $φ : G → H$ be an isomorphism of two groups. Then prove that if $G$ has a subgroup of order $n$, then $H$ has a subgroup of order $n$. Proof : For the ...
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0answers
18 views

Is a class of sets indexed by all the abelian groups a proper class [duplicate]

Let $\{K^i \mid i \in \mathbb{N}\}$ be a set indexed by $G \in \textbf{Ab}$. Is the class of all $\{K^i \mid i \in \mathbb{N}\}_{G \in \textbf{Ab}}$ a proper class? My understanding is that since ...
3
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2answers
70 views

Every isomorphism of subgroups of $\mathbb{Q}$ is of the form $\varphi(x)=qx$ for some $q \in \mathbb{Q}$.

Let $A$ and $B$ be subgroups of the additive group $\mathbb{Q}$. If $A$ is isomorphic to $B$ and $\varphi : A \rightarrow B$ an isomorphism, then show there is $q \in \mathbb{Q}$ such that ...
0
votes
1answer
25 views

Projection of a discrete subgroup of $R^n$ [duplicate]

Let $A$ be a discrete subgroup of $\Bbb R^n$ and let $V$ be a $m<n$ dimensional $\Bbb R$-subspace of $\Bbb R^n$. Is the projection of $A$ onto $V$ a discrete subgroup? I am most interested in the ...
3
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2answers
73 views

Elements of order $3$ in $\text{Aut}\left(\mathbb{Z}/91\mathbb{Z}\right)$

It looks like someone has already been here, but the question I have goes farther. To summarize my work, as well as the work in the above post, we know that ...
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1answer
32 views

Subgroups of Prüfer Group

Show that finitely generated subgroups of the Prüfer group are finite and also show that the Prüfer group is not finitely generated. I managed to prove the latter ( infinitely many primes) but I ...
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0answers
33 views

Let $ G $ is finite nilpotent group. can we say $ G $ is $ p $-supersoluble for any prime $ p $?

Let $ G $ is finite nilpotent group. Thus $ G $ is $ p $-nilpotent group for any prime $ p $. Now can we say $ G $ is $ p $-supersoluble for any prime $ p $?
3
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2answers
35 views

How to prove generator is is normal subgroup of G iff a is the center of G?

Problem: Let $G$ denote a group, and $H$ a subgroup of $G$. Suppose $a$ to be an element of $ G$ of order 2. Prove that $\langle a \rangle$ is a normal subgroup of $G$ iff a is in the center ...
3
votes
2answers
39 views

if $H \leq G$ has index 2, then $a^2\in H$ for every $a\in G$

if $H \leq G$ has index 2, then $a^2\in H$ for every $a\in G$ I am not sure that whether the way that i prove this statement is correct. Since $[G:H]=2, \forall a\in G,G/H=\{H,Ha\}$ Hence $Ha^2=H ...
1
vote
1answer
45 views

Priority of the 3 axioms of groups [duplicate]

In my book about Abstract Algebra, it is stated that A group $\langle G,*\rangle$ is a set $G$, closed under a binary operation $*$, such that these 3 axioms are satisfied: $g_1$: For all ...
0
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0answers
57 views

The trivial group, multiplication, 0 and 1 [closed]

The trivial group has one element, call it $\{e\}$. Use the binary operator for multiplication, $*$. Setting $e=1$ will work since: the group has closure: $1 * 1 = 1$ the identity element is $1$ the ...
1
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3answers
79 views

When is a given pair $(G,*)$ a group?

I'm doing practice problems for my linear algebra class, and I don't understand how to use the group axioms to see which pairs $(G,*)$ are groups. $G= (0,\infty)$ with $*$ given by addition $G= ...
1
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0answers
26 views

Intuitive argument for the transitivity of $PGL(n+1)$ acting on $\mathbb{P}^n$

In spherical geometry we can consider the action of $\operatorname{O}(n+1)$ on the unit sphere $\mathbb{S}^n$. It's easy to see that this action is transitive, because for any two points $x,y \in ...
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1answer
35 views

Can one modify the generators of a transitive group to get an intransitive group while preserving conjugacy classes?

There is a general question I'm interested in: given $g$ and $h$ with $H=\langle g,h \rangle$ a transitive subgroup of $S_n$, when is it possible to find $g',h'$ so that $H'=\langle g',h' \rangle$ is ...
0
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1answer
63 views
6
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1answer
131 views

Do uncountable groups have generating sets where you can find elements of arbitrarily long length, and what are conditions to guarantee that?

I would like to find a way to pick a set of generators in a group $G$ so that one can always find an element of $G$ of arbitrary length. I'm not sure whether or not this is always possible, and if ...
3
votes
4answers
360 views

Does group inverse commute with multiplication for all groups? [duplicate]

Is the following a property of $\textbf{all}$ groups? $a^{-1} \circ b^{-1} = (a \circ b)^{-1}$ As far as I can tell it is true for addition and multiplication, but in the notes that I have come ...
4
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2answers
360 views

Proof of Wilson's Theorem using concept of group.

I am studying group theory so I do it by using the concept of group. What I am trying to prove is if p is prime then $(p-1)!\equiv-1\mod p$ Note that $\mathbb{Z_p}$ forms a multiplicative group. ...
2
votes
1answer
27 views

Homomorphic image if smaller fails to exist

Suppose a finite group $G$ has no homomorphic image of order $n$. Is it possible for $G$ to have a homomorphic image of order a multiple of $n$? My gut says "no", as the larger homomorphic image ...
1
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2answers
43 views

Computing Factor Group step-by-step manually

I am reading Fraleigh's Abstract Algebra $\S$15 on factor group, Example #15.11: Compute factor group $(\mathbb Z_4 \times \mathbb Z_6) / \langle (2, 3)\rangle.$ The text gives a solution that ...
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0answers
27 views

$x,y \in G$, $o(x) = 2, o(y) = 2, o(xy) = k, k\geq 3$. Show that $D_k \cong \langle x,y\rangle$

I know that $D_k = \{1,r, \dots, r^{k-1}, s, sr, \dots, sr^{k-1} \}$ and that I can use $\phi: s^i r^j \mapsto x^i(xy)^j$. I don't know what can be found using this.
4
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3answers
73 views

Does the group $\mathbb{R}^{\times} / \mathbb{Q}^{\times}$ have a subgroup of order 5?

Does the group $\mathbb{R}^{\times} / \mathbb{Q}^{\times}$ have a subgroup of order 5? I don't know how I should approach this problem. Could you give me some hints on how to solve this? A more ...
1
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0answers
15 views

Branching rules without previous knowledge of the projection matrix?

Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups: $$ R= R_1 + R_2 + \ldots$$ This is often called a ...
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2answers
112 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
1
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1answer
18 views

Set of discrete orbits under subgroup of $Isom(\mathbb{E}^n)$ is clopen

I'm trying to solve the following exercise (exercise 1.4 from Szczepanski's "Geometry of Crystallographic Groups"): Let $\Gamma$ be a subgroup of $I(\mathbb{E}^n)$, the group of isometries on ...
3
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1answer
47 views

$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$

Let's consider a group $M$ (under multiplication) of all matrices $A$ of size $2 \times 2$ over $\mathbb{R}$ so that $\det(A)=1$. How to show that the group is homeomorphic to the $S^{1} \times ...
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1answer
66 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
0
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1answer
64 views

Matrix Algebras: Generator

Problem Given the algebra $\mathcal{M}_\mathbb{C}(2)$. Denote the normals: $$\mathcal{N}:=\{N\in\mathcal{M}_\mathbb{C}(2):N^*N=NN^*\}$$ And their calculus: ...
2
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0answers
52 views

Is there a simple proof of Frobenius's theorem using Sylow theory? [closed]

I ask because the proof I always stumble upon uses double induction and properties of the totient function. The statement of the theorem is: If $n$ divides the order of a finite group $G$, then ...
3
votes
1answer
173 views

False proof that simple groups are cyclic

"Proof" of not cyclic $\implies$ not simple: If $G$ is not cyclic, $\langle g \rangle$ is an abelian, and hence normal subgroup of $G$. Doesn't this show every simple group is cyclic? What am I ...
4
votes
2answers
89 views

Coproducts in $\mathsf{Grp}$

The limits and colimits in the category of abelian groups are as nice as can be, since products and equalizers are the same as in the category of sets. In the category of groups, however, the ...
0
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2answers
36 views

The intersection of all abnormal maximal subgroups of a given group

Let $G$ be a group and let $\mathrm{aFrat}(G)$ denote the intersection of all abnormal maximal subgroups of $G.$ The English summary of the paper "The intersection of abnormal maximal subgroups of ...
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2answers
35 views

Isometric group of hyperbolic 3-dim manifolds

In the book : Foundation of Hyperbolic Manifolds There is a theorem that any finite subgroup of $ Isom(\mathbb{E}^n) $ fixes a point. And I hope to solve the following question : Any ...
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votes
1answer
43 views

a question of group theory [duplicate]

let $S$ be a collection of (isomorphism classes of) group $G$ which have the property that every element of $G$ commutes only with the identity element and itself then which option is true and why ? ...
2
votes
2answers
149 views

Group Theory: group under the composition multiplication modulo $p$

Suppose you have a group $G(S,*)$ where $S=\{1,2,\ldots,p-1\}$, $p$ is prime number, and $*$ is equivalent to the multiplication$\mod p$. If $a,b$ belong to $S$, then $ab\pmod{p}$ also belongs to ...
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0answers
20 views

$ G $ is supersoluble if and only if every normal subgroup of $ G $ with index no more than 2 satisfies the maximal permutizer condition.

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
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1answer
24 views

Prove that if $H$ is a unique subgroup of order $|H|$ then $H$ is a characteristic subgroup.

Let $G$ be a group and $H\leq G$. $H$ is a unique subrgroup of order $|H|$. Prove that $H$ is a characteristic subgroup. I tried to show by contradiction that if $\phi (h)\not \in H$ then I can find ...
4
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0answers
51 views

condition for a group to be abelian [duplicate]

I’d like to prove: Let $G$ be a group and $m,n$ two relatively prime numbers. If $x^my^m=y^mx^m$ and $x^ny^n=y^nx^n$ for all $x,y\in G$, then $G$ is abelian. Thanks for help in advance.
2
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2answers
41 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
4
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3answers
125 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
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2answers
103 views

Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
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votes
1answer
49 views

True or false simple algebra questions (centralizers, conjugacy classes, normal groups, abelian groups) [closed]

Can someone please verify my answers to the following questions? Note: This is NOT homework! Answer true or false to the following questions: Two elements of a group in the same conjugacy ...
2
votes
2answers
28 views

Permutations with Condition

I have looked at this old problem in my textbook: How many permutations $\pi \in S_n (n \geq 3)$ meet the requirement: $\pi (1) < \pi (2) $ or $\pi (1) < \pi (3)$? I am not sure how to ...
2
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2answers
40 views

Equations in groups

I want to solve an equation $$f(\sigma , \tau , \delta)=1$$ where $\sigma,\tau,\delta$ are elements from a given group $G$, and $1 \in G$ is the unit element. When I say solve I mean give sufficient ...
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0answers
22 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...