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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47 views

Determining the minimum dimension required for embedding a finite group

Consider the groups $S_3$ and $S_4$ which are the symmetric groups on 3 and 4 elements respectively. We note that $S_3$ can be realized geometrically as the set of all rotations and reflections of a ...
1
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3answers
33 views

Show that if $A\cup B = A\vee B$ for subgroups $A$ and $B$, then $A\subseteq B$ or $B\subseteq A$

I have that $$A\cup B = A \vee B$$ My book defines $A \vee B$ as being: $$\cap\{T: \text{T is a subgroup of $G$ and $A\cup B \subseteq T$}\}$$ So, if I take the intersection of all subgroups ...
7
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3answers
58 views

Is there any element of order $51$ in the group $U(103)$

Does there exist an element of order $51$ in the multiplicative group $U(103)$ ? Now if the element exist say $x$ then it satisfies the equation $$x^{51}\equiv 1\pmod {103}$$ . ...
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1answer
5 views

transitive action on finite abelian subgroups

Let G be a group and K a finite subgroup of G. Let H be some subgroup of the normalizer of K in G, and assume the action of H on K by conjugation is transitive on elements of K of same order. Does H ...
-2
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1answer
25 views

transitive action on finite abelian subgroups [on hold]

Let $G$ be a group and $K$ a finite subgroup of $G$. Let $H$ be some subgroup of the normalizer of $K$ in $G$, and assume the action of $H$ on $K$ by conjugation is transitive on elements of $G$ of ...
1
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1answer
18 views

$S$ is a non empty set and there are $a$ and $b$ for $c$ and $d$ such that $a\cdot c = d$ and $c\cdot b = d$, prove it is a group

An associative operation $\cdot \ $was defined in $S$ such that $\cdot \ $is associative. Also, for all the pairs $c$ and $d$, there are elements $a$ and $b$ such that: $$a\cdot c = d, \ \ \ \ c\cdot ...
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0answers
40 views

How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
0
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1answer
26 views

Verifying if these Cayley tables are from groups

For the first table I noticed that $ab = c \implies abb = cb \implies a = cb$ but in the table, $cb = d$, so this can't be a group For the second table, we have: $ab = c \implies (aa)b = ac ...
-4
votes
1answer
32 views

Two sub-groups of order 3 and 5, prove that the group of order 15 is cylic. [on hold]

So all I have is that G is a group of order 15, and there are 2 unique sub-groups, which order is 3 and 5 (I mean there only one sub-group of each kind) and I need to prove that G is cyclic. Dont see ...
3
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0answers
27 views

Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then ...
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0answers
38 views

$G$ of finite order $2p$ ($p$ is prime). Prove that $G$ abelian. [duplicate]

I have a group $G$ of order $2p$, where $p>2$ and prime. The additional thing that I also know, that $\exists a\in Z(G)\mid O(a)=2$. I need to prove that G is abelian. But first, before that, ...
3
votes
0answers
41 views

Standard notation for indices in group theory?

I've seen three notations for indices in group theory, namely $(G:H)$, $[G:H]$ and $|G:H|$. Is there any of these notations that is standard?
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0answers
21 views

Finitely generated abelian groups and finite index subgroups

I want a proof or reference for the following fact: "Let $G$ be a finitely generated abelian group and let $\phi:G\to G$ be an injective homomorphism. Then the index $[G:\phi(G)]$ is finite." I ...
0
votes
1answer
37 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
2
votes
2answers
36 views

Proof that normalizer and center are subgroups

I've seen this proof for the center of a group $G$: $$C = \{x\in G:xg = gx \ \ \ \forall g \in G\}$$ So, the center is the set of all elements that commute with every $g$ of $G$. This subset of $G$ ...
1
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1answer
49 views

Discontinuous $ f : \mathbb R^2 \to \mathbb R$ with unusual topology on $ \mathbb R$

With the usual topology on the reals $\mathbb R$ , let $D$ be the family of dense open sets and let $T=D \cup \{ \phi \}$. Let $S$ be the set $R$ with the topology $T$ on it. Show that the function ...
3
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1answer
82 views

Finiteness of subgroup $\rightarrow$ Finiteness of the group

Let $G$ be a group and $H$ be its abelian and normal subgroup. If $H$ is finite and maximal, prove that $G$ is finite. What I tried : Assume $H=\{e,h_2,\cdots,h_{n}\}$. As for each $j$, we ...
1
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1answer
23 views

Index of every maximal subgroup is prime number

Suppose that finite group $G\neq 1$ and $|G : M| ∈ \mathbb P$ for every maximal subgroup $M$ of $G$. Then prove: $G$ contains a normal maximal subgroup. (we all know that a maximal subgroup is normal ...
2
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0answers
34 views

Determine $\langle x,y: x^2=y^2, xyx=y \rangle$

We have a finitely presented group $G = \langle x,y: x^2=y^2, xyx=y \rangle$. It is easily shown the orders $|x|,|y|$ of $x,y$ are divisors of $4$, with $|x| = 4 \Leftrightarrow |y|=4$ and $|x| \leq 2 ...
0
votes
0answers
68 views

$(\mathbb Z /n\mathbb Z)^*$ is cyclic for all $n$ implies every finite abelian group is cyclic. [on hold]

Show that, if $(\mathbb Z / n\mathbb Z)^*$ is cyclic for all $n \ge 1$ then every finite abelian group is cyclic. More precisely, prove that the following are equivalent: 1) $(\mathbb Z / n\mathbb ...
1
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1answer
19 views

central series of $\frac{G}{Z_2(G)}$

Let $G$ be finite p-group I am trying to make central series for $\frac{G}{Z_2(G)}$ and more inportant what is nilpotency class of $\frac{G}{Z_2(G)}$ since ...
2
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1answer
34 views

Presentation of Abelianization of a group

Say $G$ is a finite group with presentation $\langle S | R \rangle$ and let $C$ be the commutator subgroup of $G$. Then $\langle S | R \cup \{ sts^{-1}t^{-1} \} \rangle$ is a presentation of $G/C$. ...
2
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1answer
61 views

Set of all inner automorphisms is a normal subgroup

In order to prove this, I first proved that the set of all automorphisms from a group $G$ to $G$ form a group under composition: The identity homorphism is an automorphism because sends $x$ from $G$ ...
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votes
0answers
23 views

A group with max-n is an FC group

A group with max-n (i.e maximal condition on normal subgroups) is an FC-group (i.e a group in which every conjugacy class is finite) if and only if it is group whose center has finite index. I can't ...
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votes
0answers
31 views

assume $ M/N $ be a chief factor of $ G $. Why $ M/N $ has prime order or order $ 4 $?

Let $ G $ is a soluble group and $ \Phi(G) $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal ...
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0answers
21 views

minimal normal subgroup of a chief factor of soluble group $ G $ is a minimal normal normal subgroup of $ G $?

Let $ G $ is a soluble group and $ \Phi(G) $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent and $ M \neq 1 $. assume $ M/N $ ...
0
votes
1answer
55 views

need examples of different groups

I need example of different groups having different properties like: class 2 or 3 cyclic commutator cyclic center $Z(G)\le \Phi(G)$ redei group $G=\langle aG',bG' \rangle $ and ... Is there books or ...
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1answer
39 views

Order of a permutation group versus degree of a permutation group

Excuse my simple question. I am just starting to learn about group theory. I am trying to understand the description of cycle index for a permutation group. The Wikipedia entry references both the ...
0
votes
1answer
50 views

Endomorphisms of $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$

Let G be any abelian group, End(G) be the set of all group homomorphisms $\varphi\colon G\to G $. End(G) is a unital ring under the operations + and $\cdot$(Please refer to the link for detail, ...
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0answers
43 views

abelian subgroups of $GL(2,\mathbb{Z}_p)$

Is there a classification of abelian subgroups of $GL(2,\mathbb{C})$? or $GL(2,\mathbb{Z}_p)$? Here $\mathbb{Z}_p$ is the ring of $p$-adic integers.
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20 views

Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent

Let $ G $ is soluble group with $ \Phi(G) = 1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal ...
4
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0answers
19 views

Haar measure on a profinite group is the inverse limit of the counting measure on its quotients?

I've heard this a few times now, though I've never seen a precise result. I guess the precise statement would be close to: Let $N_i$ be a basis normal subgroup neighborhoods of the identity in a ...
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2answers
39 views

Decomposition of an unitary operator by simple operators

For quantum computation, it's well known that any unitary operator can be approximated with an arbitrary accuracy by simple operators, for example to approximate an unitary operator on n qubits by no ...
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0answers
17 views

IS $ F(G) $ a direct product of some minimal normal subgroups of G?

Let $ G $ is a finite group and $ F(G) $ is the fitting subgroup of $ G $. IS $ F(G) $ a direct product of some minimal normal subgroups of G? Why ? $ F(G) $ is the largest nilpotent normal subgroup ...
6
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2answers
73 views

Is $G/H$ always a subgroup of $G$?

Given a normal subgroup $H$ of a finite group $G$, is there always an injective homomorphism $$\varphi:G/H\to G?$$ In other words, is $G/H$ a subgroup of $G$? If we pick an arbitrary ...
1
vote
1answer
15 views

on finitely generated non-abelian p-group

Let $G$ be a finitely generated non-abelian p-group for example $G=\langle x,y,z\rangle$ is following argument right? that mean for every $g\in G$ there is i,j,k $g=x^iy^jz^k$ since G is non abelian ...
1
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1answer
31 views

Let $ H/N $ is nilpotent quotient subgroup. Then is $ N $ characteristic in $ H $? If no, then what condition need?

Let $ G $ is finite solvable group and $ H $ is normal subgroup. Let $ H/N $ is nilpotent quotient subgroup. Then is $ N $ characteristic in $ H $? If no, then what condition need?
3
votes
1answer
52 views

Action of ${\rm Aut}(G)$ on $G$

Let $G$ be a finite group and consider the natural action of ${\rm Aut}(G)$ on $G$ and let there are two orbits under this action. How could we show that $G$ is an (elementary) abelian group? Is ...
2
votes
1answer
25 views

Problems with P. Hall theorem proof (The problem involves the use of Frattini's argument)

Theorem Let G be a solvable group of order $ab$, where $(a,b)=1$. Then $G$ contains at least one subgroup of order $a$, and any two such are conjugate. Details The proof the book presents involves ...
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1answer
37 views

Find the class equation for the following groups

Can someone please verify these? I'm quite unsure about my answer to the Quaternion Group. Find the class equation for the following groups: (a) The Quaternion group (b) $D_5$ (c) ...
2
votes
1answer
21 views

How many subgroups are there in an elementary-$p$ group

$C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally. But how many subgroups in $C_p\times C_p \times C_p\times C_p$? ...
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1answer
73 views

Cayley's theorem

As according to Cayley's theorem "Every group is isomorphic to a subgroup of some symmetric group". Now my question is: the additive group of real numbers is isomorphic to which permutation group... ...
3
votes
1answer
35 views

Find the order of the conjugacy class of $A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ in $GL_2(\mathbb{F}_5)$

Can someone please verify my answer? Note: This is not homework, only self study. Find the order of the conjugacy class of $A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ in ...
4
votes
0answers
57 views

Is the monster group a characteristic quotient of $F_2$?

Let $F_2$ be the free group on two generators, and $M$ the monster group. It's known that every finite simple group is 2-generated, so let $F_2\rightarrow M$ be a surjection with kernel $N$. Let ...
0
votes
1answer
52 views

Let $G$ be a group such that $|G| \ge 3$ then $|AutG| \ge 2$. [duplicate]

Let $G$ be a group such that $|G| \ge 3$ then $|AutG| \ge 2$. How can I approach to this problem? It is necessary to divide in cases? For G finite and infinite, or Abelian and non-Abelian? The ...
2
votes
2answers
90 views

Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?

Is easy to show that $(\mathbb{R},\mathbb{Q},+,\cdot)$ is isomorphic to $(\mathbb{R}^n,\mathbb{Q},+,\cdot)$ as vectorial spaces and then $(\mathbb{R},+)$ isomorphic to $(\mathbb{R}^n,+)$. This result ...
0
votes
1answer
30 views

A group of order 2p (p prime) and other conditions - prove abelian.

I have G where $|G|=2p$ ; p is prime. $\exists a\in Z\left(G\right);\:a^2=e$. I need to prove that G is abelian. Now, let's translate it into math. To prove that G is abelian, is in other words ti ...
3
votes
3answers
33 views

If a group $ G$ is not simple does it follow that it is isomorphic to the direct product of two nontrivial groups?

Let $G$ be not simple does it follow that $G=G_1\times G_2$, where $G_1$ and $G_2$ are nontrivial groups? Edit: Wait is the answer $G_1=N\lhd G$ and $G_2=G/N$?
0
votes
1answer
23 views

Can non-cyclic one-relator groups be finite?

The answer appears to be no, but I can't find it anywhere. Worded another way, do there exist subgroups of finite index in the free group $F_2$ on two generators $x,y$ which is normally generated by ...
5
votes
1answer
75 views

If groups $G$ and $H$ act on $X$, does $G\times H$ act on $X$?

Suppose two groups $G$ and $H$ act on a set $X$. What is the a group action of $G\times H$ on $X$? From the actions there a homomorphisms $\varphi\colon G\to S_X$ and $\psi\colon H\to S_X$. So this ...