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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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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23 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 ...
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1answer
40 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
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2answers
39 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
84 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 ...
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36 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 ...
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1answer
21 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
62 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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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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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
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1answer
57 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
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1answer
51 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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0answers
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
20 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
41 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
77 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
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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 ...
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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
42 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
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1answer
22 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
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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
55 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
91 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 $G_2=G/N$?
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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
77 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 ...
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0answers
33 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...
6
votes
1answer
101 views

Space of arbitrary rotations of a cube

Suppose I have a cube $[-1,1]^3\subset\mathbb{R}^3$. I am allowed to rotate it about any angle/axis through the origin rather than just $90^\circ$ about the coordinate axes, e.g., by applying ...
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1answer
41 views

p-element centralizing a Sylow p-subgroup

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and $g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$. Is this true? How can i prove that $g \in P$ ...
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1answer
38 views

characteristic of a ring

I got 4 short question about characteristic. 1) What is characteristic of integral domain D which suffices $20 \cdot 1_D=0_D=12 \cdot 1_D$ 2) Let $A=\{0,1,a\}$ be a integral domain what is ...
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1answer
99 views

How many non isomorphic semidirect products are there between $\mathbb Z_2$ and $SL(2,3)$?

I know that $GL(2,3)$ is one of this, but i need the characterization of all possibles of the semidirect products between $\mathbb Z_2$ and $SL(2,3)$. Thanks, for any help.
0
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1answer
25 views

difference between p-automorphism and automorphism of order p

Is there difference between p-automorphism and automorphism of order p? I looked up in internet some time they say they are same but some time it's look like there is a different between them for ...
2
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0answers
51 views

I have a finite permutation group and access to a computer algebra system, how can I recognize the structure of the group?

I have a fixed permutation group $G$, and cannot tell which finite group it really is in a "human readable" way. I also have GAP. Is there a step by step computation to give me the structure of this ...
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1answer
43 views

Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent. What's mean smallest normal subgroup?

theorem: Let $ G $ be solvable with $ \Phi(G)=1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Then every chief factor of $ G $ has prime order or is $ G $-isomorphic ...
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1answer
28 views

how many elements are there which has order 3?

if the number of sylow-3 subgroups of a group of order 96 is 4, how many element are there which has order 3? I dont know how to start. why isnt the answer 4?
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1answer
39 views

Smallest normal subgroup and minimal normal subgroup, what's the difference?

Let $ G $ is a finite group and $ N $ be a minimal normal subgroup of $ G $ and $ M $ be a smallest normal subgroup of $ G $. Smallest normal subgroup and minimal normal subgroup, what's the ...
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2answers
58 views

which one of these may not be abelian?

If a group G has these orders. which one of these may not be abelian? 4,31,55,39 and 121 since 4 and 121 is prime square. they are abelian. and 31 is prime therefore cyclic so abelian. what about 55 ...
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27 views

Center of GL(2,F) and SL(2,F)

My question is to find the center of GL(2,F) and SL(2,F) where $F=\mathbb{Z},\mathbb{C}$ My attempt: Generalising the identity element, if we take $\pmatrix{ a& 0\\\ 0& a}$. Then these ...