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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find all subgroups of G where: $0 \ne r \in \Re$ $G = <r>$

I need to find all subgroups of G where: $G \lt \Re$ $0 \ne r \in \Re$ $G = <r>$ $\Re$ is the group of real numbers and G is a subgroup. Edit : the operation is + I tried thinking about ...
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35 views

Possible order of $ab$ when order of $a$ and $b$ are known.

Let $a,b\in G$ be elements of a finite group $G$. We know $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. In dependence of $m$ and $n$ what are the possible values of ...
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Given a group of order $p^nq^2$ for two odd primes, prove that the commutator is a p group.

Given a group of order $p^nq^2$ for two odd primes $p > q$, prove that the commutator is a p group. To solve this question I need to prove that the commutator can't be of the orders $p^iq$, ...
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commutator (derived) subgroup of S3

how can i calculate it easily? i showed that the commutator group of S3 is generated by (123) in S3 using the fact that S3 is isomorphic to D6 and relation in D6 but that was tedious...are there any ...
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subgroups of the free product of a finite cyclic group and an infinite cyclic group.

How can I find all the subgroups of the free product $G = \mathbb{Z}*\mathbb{Z_2}$? I tried to answer this by looking at the subgroups of $\mathbb{Z}$ and $\mathbb{Z_2}$ separately. The subgroups of ...
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1answer
18 views

What should I do to tackle the following matrices calculation?

Through chapter 3 of Group Theory by Morton Hamermesh in part 3-6 (Equivalent representations; characters.) I stopped in some point. It's told "If we change the basis in the n-dimensional space $L$, ...
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Find the bigger possible order of element in the group $Z_2 \times Z_{36} \times Z_{10}$.Give an element in the group that has the order we found

Find the bigger possible order of element in the group $Z_2 \times Z_{36} \times Z_{10}$.Give an element in the group that has the order we found. How i can find the bigger order? i saw an example ...
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Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic

Let $G$ group of order $pq$ where $p,q$ primes.Show that if $G$ contains normal groups $N$ and $K$ with $|N|=p$ and $|K|=q$ then is cyclic Any ideas or hints for showing this?
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21 views

calculation $p$-Fitting subgroup

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $ that $ \vert A \vert = p^{a} $, $ p $ is prime. Let $ N =Fit(G) $, then $ N = O_{p}(G) $. Suppose $ F/N = ...
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1answer
35 views

Does such homomorphism exist?

$G$ is a group: $|G|=20$. Is there such a group G, for which the homomorphism $\tau :G-->Z_{10}$ exist?$$$$ The same question for: $\tau :G-->Z_{15}$ $$$$ I think that I should use here the ...
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2answers
33 views

Capable groups of order $32$ with GAP

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. How can I find all capable groups $G$ of order 32 with $|Cent(G)|=10$, where $Cent(G)$ is the set of all ...
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1answer
15 views

Subgroups of the unit circle under complex multiplication

Show that there are different subgroups of the unit circle which are isomorphic to ZxZ. I can show there are many subgroups of the unit circle which are Isomorphic to Z but I am having no idea for ...
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1answer
16 views

How to describe the transformation that changes French flag to Russian flag?

http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=Russia%2C%20France%20flags I presume it can be described two group operators, but I'm not sure how to come up with the formal description. ...
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1answer
11 views

Normalizer of Unipotent subgroup in General Linear group

Let $\mathrm{GL}(n,\mathbb{F}_p)$ be the general linear group over field of order $p$, and $\mathrm{U}(n,\mathbb{F}_p)$ be the subgroup consisting of upper triangular matrices with each diagonal entry ...
2
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1answer
40 views

When the elements of maximum order are $n$-cycles in $S_n$?

If the elements of maximum order in $S_n$ are $n$-cycles, then we can guess with few computations that $n$ must be at most $4$. How can we prove this? I tried the case in $S_{2n+1}$, the symmetric ...
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1answer
31 views

The concept of parity for members in a group

I was wondering if the concept of an even number has a construction within group theory? Furthermore does it have any application or further abstraction? For example; as we know that all even number ...
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1answer
31 views

$x$ and $g$ are elements of the group $G$, show that the order of $x$ is equal to the order of $g^{-1} xg$.

If $x$ and $g$ are elements of the group $G$, prove that $|x| = | g^{-1} xg|$. Deduce that $|ab| = |ba|$ for all $a,b \in G$. attempt: Let $|x| = n$ be the order of $x$ and $| g^{-1} xg| = m$ be the ...
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23 views

How many and which homomorphisms are there from $S_3$ to $Z_8$? After this find all the possible automorphisms of $Z_9$ [duplicate]

How many and which homomorphisms there are from $S_3$ to $Z_8$?After this find all the possible automorphisms of $Z_9$ Any ideas or help for finding this homomorphisms and automorphisms?
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32 views

Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
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Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ ...
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For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
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Infinitely iterated square roots in groups

Let $G$ be a group. What are possible conditions on $G$ to ensure that there is no sequence $\{g_i\}_{i\in\mathbb Z}\subset G\backslash\{1\}$ such that $g_{i+1}=g_i^2$ for all $i\in\mathbb Z$? Does ...
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1answer
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Cyclic Groups - $a^k = e \text{ iff } n|k$

I saw this proof in the book on Abstract Algebra. Here is part of it: Let $G$ be a cyclic group of order $n$ and $a$ is the generator of $G$. Then $a^k = e \iff n|k$ Proof: Suppose $a^k=e$. By the ...
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28 views

Prove that the center of a group with $385$ elements has an element of order 7. [duplicate]

Prove that the center of a group with $385$ elements has an element of order 7. By Cauchy's theorem I know that if I prove that $Z(G)$'s order is divisable by 7, we're done. So now I need to rule out ...
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14 views

The conjugate closure of a subset is a kernel of a permutation representation associated to a group action

Let $G$ be a group and $H$ be a subgroup of $G$. Let $A$ be the set of all left cosets of $H$ in $G$. We know that $G$ can acts on $A$ by $$g\cdot xH=gxH.$$ For any $g\in G$, define that ...
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21 views

Application of Bruhat decomposition in $\mathrm{GL}(n,\mathbb{F}_p)$

Consider the General Linear group $\mathrm{GL}(n,\mathbb{F}_p)$ over the field of order $p$, $B$ denote the subgroup consisting of upper triangular matrices, and $W$ denotes the subgroup consisting ...
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1answer
39 views

How to prove that O(Ng) | O(g)

I have this exercise: Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all $Ng\in G/N$, $$o(Ng)\mid o(g).$$ For now, without using the canonic homomorphism $\tau ...
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40 views

Homomorphisms from $\mathbb{Z}_p$ to $\mathbb{Z}_3$

For which odd values of $p$ can we find a non-trivial homomorphism from $\mathbb{Z}_p$ to $\mathbb{Z}_3$ ? Is there any method to find those homeomorphisms explicitly? I have no any idea to handle ...
2
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2answers
160 views

Group Action as permutations

I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . then the action may be viewed as permutation by $x ...
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30 views

Condition under which $HK$ is a subgroup

Suppose $G$ is a finite group and $H$, $K$ are subgroups. $H < N_G(K)$ is a sufficient condition for $HK$ to be a subgroup, but is is possible that $HK$ is a subgroup although neither $H$ nor $K$ ...
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1answer
36 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
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1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
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Enumerating functions modulo action on both the domain and codomain.

Let $Hom(A,B)$ be the set of functions from a finite set $A$ to a finite set $B$ and let $G_A \leq S_A$, $G_B \leq S_B$ be a subgroups of the permutation groups of $A$ and $B$. For $f,g \in Hom(A,B)$ ...
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27 views

Third isomorphism theorem, about quotients

I'm trying to understand the third isomorphism theorem statement, specifically the one in Wikipedia: https://en.wikipedia.org/wiki/Isomorphism_theorem#Third_isomorphism_theorem I'm stuck at the point ...
3
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32 views

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...
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Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $.

Let $ G $ is a soluble group and $ N $ minimal normal subgroup of $ G $. Let $ F = Fit(G) $. Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $. I show ...
5
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1answer
32 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
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1answer
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prove unique minimal normal subgroup of soluble group by $ P_{G}(M) > M $

Let $ G $ be a soluble group. If $ P_{G}(M) = \langle y\in G | \langle y \rangle M = M\langle y \rangle \rangle > M $for any subgroup $ M $ of prime power index in $ G $, then every chief factor ...
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$\mathbb{C}\{X\}^\chi$ a $G$-stable subspace of $\mathbb{C}\{X\}$?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = SL_2(\mathbb{F})$. The group $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group ...
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1answer
55 views

Commutative Diagram for group structure

I remember seeing once a commutative diagram that explained group structure. Where the associativity, identity element, inverse, multiplication and all was shown in a singular diagram, it is trivial ...
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1answer
16 views

Torsion-free nilpotent Group

I am looking for a short proof to the following fact: In torsion-free nilpotent group we have: an non-trivial element cannot be conjgate to it inverse. I know very little about nilpotent ...
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How to find a Sylow p-group?

I use Mathematica to do what I refer to as computational group theory. I know how, theoretically, to find a Sylow p-group. I want to know how to use Mathematica to solve this: Given a (large, finite, ...
3
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$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h ...
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Decomposition of certain representation of cyclic group by irreducible.

Let $C_{n}$ denotes cyclic group of order $n$. Let the set of real irreducible representations of $C_{n}$ can be listed as $$ \begin{cases} \{1,\xi,\xi^2 , \cdots \xi^{(n-1)/2}\}, \text{when $n$ is ...
2
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1answer
38 views

Why does there exist a deck transformation mapping here?

See Kevin Dong's answer here. Let $\widetilde{\gamma}$ be a path in $X_N$ based at $x_0^N \in X_N$ with $p(\widetilde{\gamma}) = \gamma$ a nontrivial path in $X$. Since the cover is normal, there ...
4
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1answer
58 views

Is there a “unique factorization theorem” for finite groups?

Sometimes it is difficult for me to understand what a group seems like. For example, the dihedral group $D_5$ is easy to visualise when I think it of as a "product" of two cyclic groups $C_2$ and ...
3
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1answer
37 views

$ S_{4} $ has a sylow tower?

A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, ...
4
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1answer
60 views

If $|H|=112$ then $A_7\cap H \lhd H$?

I posted this because Alex Clark asked in chat and I'm not sure how to proceed. Let $G$ be a group such that it has a fixed subgroup isomorphic to $A_7$, which we denote simply by $A_7$. Let $H$ be a ...
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1answer
48 views

On formula $\sum_{i=1}^n 1/(G : H_i) = 1$ on a group $G$

Let $G$ be a group. Let $H$ be a subgroup of $G$ such that $(G : H) \lt \infty$. Then there exists a sequence of elements $a_1,\cdots, a_n$ such that $G = \bigcup_{i=1}^n a_iH$ is a disjoint union. ...
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45 views

Is there a link between music theory and the mechanics of the universe? [on hold]

The production(formation)[death] of a chromatic(spherical)[gravitational] piece(droplet)[star] of music(liquid)[space/time] minimizes the tonal-area(surface-area)[dimensions] which is the ...