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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GCF of Fractions whose Numerators and Denominators are Co-Prime

Suppose I have a set of rational numbers where elements have denominators are odd and numerators and denominators are co-prime. I need to show that the set is closed under addition. It is clear that ...
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18 views

$Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ .

Let $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . a) Show that $v$ commutes with $u^3$. [Show that $v^2u^3v = u^3$ by writing the ...
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Supersolubility of $ G/H $ and $ H $ not deduced supersolubility of $ G $.

I want show $ S_{4} $ isn't supersoluble group. For this suppose $ 1 \leq B_{4} \leq A_{4} \leq S_{4} $ be a normal serie of $ S_{4} $, that $ B_{4} $ is Klein’s four-group. Since $ B_{4} $ isn't ...
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12 views

Difference between 1 (usual) and 1 bar of cayley table?

Why we write 1 as 1 bar in cayley table, instead of usual 1.
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2answers
26 views

What is the center of $\mathbb{C}S_3$?

How do I found the center of symmetric group algebra $\mathbb{C}S_3$? and in general $\mathbb{C}S_n$? I did an example on a smaller group algebra: $\mathbb{C}S_2=\{a (1)+b(12) \mid a,b\in ...
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1answer
9 views

$f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$?

Let $f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then is it true that $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$ ? Since both $f,g$ are non-trivial , I ...
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35 views

Question about sub-groups.

Let H,K be sub-groups of G (finite order), proof that if (G;H) and (G:K) are relatively Prime G=HK. Any clue?
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1answer
30 views

Chosen maximal subject is a subgroup

Let $ G $ is a finite soluble group and $ N $ be a unique minimal normal subgroup of $ G $. Let $ G = TS $ that $ S $ is the fitting subgroup of $ G $ and $ T = N_{G}(H) $ for $ H \leq G $. Suppose $ ...
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26 views

Abelian Sylow $p$-subgroups of group $G$. [on hold]

Let $G$ be a non-Abelian group. prove that if $G/Z(G)$ is isomorphic to alternating group $A_4$, then every Sylow $p$-subgroup of $G$ is Abelian.
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1answer
32 views

If two elements commute, does each element commutes with the inverse of the other [on hold]

Let $G$ be a group and $u,v \in G$. Is it possible that $uv = vu$ but $u^{-1}v \ne vu^{-1}$?
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2answers
24 views

Sylow p-subgroup of order p does not normalize any other Sylow p-subgroup

Let $P_1,P_2$ be distinct Sylow p-subgroups of $G$ with order $p$. Is it generally true that $P_1$ cannot normalize $P_2$? I've seen algebra textbooks use this fact for $p=3,11$ and they quote 'a ...
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0answers
32 views

The order of the derived group [on hold]

Let $G$ be a non-Abelian group. Prove that if $G/Z(G)$ is isomorphic to the dihedral group $D_8$, then $|G′|=4$.
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1answer
30 views

Subgroups of generalized dihedral groups

A generalized dihedral group, $D(H) := H \rtimes C_2$, is the semi-direct product of an abelian group $H$ with a cyclic group of order $2$, where $C_2$ acts on $H$ by inverting elements. I know that ...
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0answers
18 views

Extension of group

Let $ p = 2 $, and Let $ H $ be a subgroup of a direct product of copies of $ S_{3} $. Why $ H $ is an extension of a $ 3 $-group by a $ 2 $-group, and $ H/O_{p^{\prime}}(H) $ is a $ 2 $-group ?
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14 views

Number of sets containing m decomposable permutations of n objects.

Let $P_{m,n} = \{ \sigma_i \in S_n \}$ be a set containing $m$ arbitrary permutations of $n$ objects. Let $Q_{m,n} = \{\sigma_{ij} = \sigma_i^{-1}\sigma_j \mid \sigma_i, \sigma_j \in P_{m,n} \}$ be ...
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3answers
29 views

A more formal but intuitive understanding (on a definition) of a group action

We know that a symmetric group $S_n$ acts on the set $\{1, 2,\ldots, n\}$. The definition of an action of a group $G$ on a set $S$ is a function $G\times S\to S$ such that: 1) $e\ast s=s$ 2) ...
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1answer
32 views

Find isomorphism between $S_3$ and $GL_2(F_2)$. [duplicate]

Find isomorphism between $S_3$ and $GL_2(F_2)$. proof: Let $A = \begin{pmatrix} a& b\\ c & d \end{pmatrix}$. Where $\det (A) \neq 0$. And recall $S_3$ is the permutation group with ...
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1answer
23 views

Question in line of proof for first isomorphism theorem

Let $\phi: G_1 \to G_2$ be a group homomorphism. Let $\ker \left({\phi}\right)$ be the kernel of $\phi$. Then: $\operatorname {Im} \left({\phi}\right) \cong G_1 / \ker ...
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2answers
56 views

Is $S_5$ isomorphic with the direct product $A_5 \times Z_2$? [on hold]

Is $S_5$ isomorphic with the direct product $A_5 \times Z_2$? How i can check it?
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1answer
41 views

Actions of a finite group.

I've been playing around with this proof for a while and I can't seem to figure out where to go from here: I have that $M$ is a manifold, $G$ is a finite group (say, of order $n$), and the action of ...
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2answers
41 views

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$

Let $f:Z \times Z \to Z$ with $f(1,1)=2$ and $f(3,5)=6$. Estimate the $\ker f$ of $f$ and $f(0,5)$. I am trying to solve this but i need any ideas or hints to start,any help would be interesting.
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26 views

Connectedness of automorphism group of a variety

Let $Y$ be a proper, smooth, integral variety over an algebraically closed field $k$ of characteristic zero. Consider the automorphism group $Aut_k(Y)$ (a group scheme). Are there any natural ...
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41 views

The standard representation of $SO(n)$

This fact always bothers me. The group $SO(n)$ is defined as a set of $n\times n $ matrices, so this is a representation of the group in the first place. Yet, it seems that this representation has not ...
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1answer
31 views

If $H<G$ is a normal subgroup of G,Show that the center $Z(H)$ of H is also normal subgroup of $G$ [on hold]

If $H<G$ is a normal subgroup of G,Show that the center $Z(H)$ of H is also normal subgroup of $G$ Any ideas for showing this?
2
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1answer
18 views

$ G/N $ is is a subgroup of a direct product of copies of the cyclic group $ C_{p-1} $ if $ p>2 $

Let $ G $ is soluble group and $ A $ be a unique minimal normal subgroup of $ G $. Then $ A $ is a elementary abelian group of a prime power. Let every chief factor of $ G/A $ has order $ 4 $ or a ...
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2answers
22 views

How to show #Hom$(C_a, G)=\{x\in G: x^a=e\}$?

I am willing to establish that $$\#\text{Hom}(C_a, G)=\#\{x\in G: x^a=e\}$$ where $G$ is finite group of order $n$ and $C_a$ is cyclic group of order $a$. I started like this: By first isomorphism ...
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4answers
59 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...
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16 views

The Second Homomorphism Theorem for groups

$G$ is a group. $N,H\le G$ $$$$ N is a normal sub-group of G. Want to prove $NH\backslash N\:$ isomorphic to $H\backslash N\cap H$ without using any homomorphism, but with understanding the quotient ...
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0answers
14 views

when a dihedral groups is nilpotent?

well i found this question answered in Is the dihedral group $D_n$ nilpotent? solvable? but the answer involves the sylow thm which i haven't studied. i'm studying bhattacharya and this question ...
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1answer
22 views

If $G_1/N \unlhd G/N$ then $G_1 \unlhd G$?

I want to show that if $N$ is a simple normal subgroup of a group $G$ such that $G/N$ has a composition series, then also $G$ has a composition series. I think I can finish the proof if I can only ...
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9 views

Let $G=Z_6 \times Z_9$ with $H=<(3,3)>$ and $K=<(0,1)>$. Find the order of $(1,0)$ in $G/H$ and in $G/K$

Let $G=Z_6 \times Z_9$ with $H=<(3,3)>$ and $K=<(0,1)>$. Find the order of $(1,0)$ in $G/H$ and in $G/K$ and determine the groups:$H \cap K$,$G/H$,$G/K$,$HK/K$,$HK/H$ and $G/H \cap K/H/H ...
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1answer
20 views

Order of an element in a finite group - what does it tell me? [on hold]

I get the basics on how to calculate the order of an element in a finite group, but not sure why I want to. When I find the order what does it allow me to do or know about the group? Real world ...
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2answers
30 views

A group $G$ of order $32$ act in a set $X$ order $15$.Show that there is at least one element in set $X$ that remains stable under the action of $G$ [on hold]

A group $G$ of order $32$ act in a set $X$ order $15$.Show that there is at least one element in set $X$ that remains stable under the action of $G$ Any ideas and hints to show this?
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1answer
28 views

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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1answer
56 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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1answer
40 views

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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2answers
25 views

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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1answer
24 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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2answers
19 views

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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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
38 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
40 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
16 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
21 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
14 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 ...
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1answer
42 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
36 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
40 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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0answers
24 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?