The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...
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56 views

Pushout of a subgroup

Let $G$ be a group, $H\subseteq G$ be a subgroup and $\iota: H\to G$ the inclusion map. What is the "pushout along $\iota$? How is it constructed?
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46 views

Show that $Z(θ^G)≤H$

Suppose $H ≤ G$ and $θ \in Char(H)$. Show that $Z(θ^G)≤H$. ($Z$ is the centre and $θ^G$ is the character induced by $G$)
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37 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
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27 views

What is the $l$ in this group? [closed]

What is the $l$ in this group? $a^7=e$, $b^3=e$, $b^{-1}ab=a^2$, $ba=a^lb$. $l=?$
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17 views

Example of $1-1$ Correspondence with Subgroups of Factor Group

I am working out an example to deomstrate the one-one correspondence between $\{\text{subgroups of}\ D_4/N\}$ and $\{\text{subgroups of $D_4$ that contain $N$}\}$ but I am short one in $D_4$. ...
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1answer
47 views

Sylow's theorem and uniqunes of normal supgroup

Let $G$ be a finite group of order $pq,$ where $p$ and $q$ are primes such that $p < q.$ Then how to prove that $G$ has a unique normal subgroup of order $q?$
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35 views

Number of $n$-cycles fixed by a permutation of $S_n$.

Let $\sigma\in S_n$ be a permutation consisting of $m$ $d$-cycles. I want to show that the number of $n$-cycles $\sigma$ commutes with is $\phi(d)m^{d-1}(d-1)!$. If I write $\sigma=(a_{1, 1},\ldots, ...
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1answer
14 views

Symmetric Group and its centralisers

https://www.dropbox.com/s/te4uxryk82h9b23/Screenshot%202015-04-16%2000.38.42.png?dl=0 Let $S_7$ be the symmetric group of degree 7 (the group of permutations of the set {${1, 2,...,7}$}. Let ...
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1answer
33 views

Show that the alternating group $A_9$ has no subgroups of index 8?

So far, I believe it's a proof by contradiction. Suppose that $H \leq A_9$ with $[A_9 : H] = 8$.. $|H| = |A_9|*8$(which is a large number)? then would this involve the 3-cycles? Quite stumped. Thank ...
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34 views

Suppose K:Q is normal with Gal(K:Q) isomorphic to $\mathbb{Z}_4\times\mathbb{Z}_6$. Prove that K has 3 distinct non-trivial square roots.

Suppose K:Q is normal with Gal(K:Q) isomorphic to $\mathbb{Z}_4\times\mathbb{Z}_6$. Prove that K has 3 distinct non-trivial square roots. Q is the set of rational numbers. The only clue that I've ...
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45 views

automorphism group of direct product of groups

I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups. Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then ...
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25 views

Symmetrical group on 3 Elements

Let S3 be the symmetric group on three elements. Find all normal subgroups of S3. I have the following: Lagrangeís theorem implies that the subgroups of S3 = Aut(fa; b; cg) are of orders 6; 3; 2; 1: ...
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65 views

How do I show that $\mathbb{Z}_4\times\mathbb{Z}_6$ has only 3 subgroups of size 12?

I am aware of what the 3 subgroups are, and now I am trying to show that these are the only 3 subgroups of size 12. I am trying to work with their indexes being 2, implying that they are normal, but ...
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37 views

Since an automorphism σ is a bijection, its inverse σ inverse is well defined. show that σ inverse is an automorphism of g?

An Isomorphism of G onto itself is called an automorphism of G. A)Since an automorphism σ is a bijection, its inverse σ inverse is well defined. show that σ inverse is an automorphism of g? B) Show ...
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33 views

Subgroup of a cyclic finite group [closed]

Let $G$ be a cyclic group of order $n$ and let $m$ be a positive integer dividing $n$. Show that $G$ contains one and only subgroup of order $m$. I have started the proof by stating the smallest ...
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12 views

Where does $b(10)$ goes under this automaton

This is finite state automaton for Grigorchuk group. I have never studied automaton formally, so I wanna check is it fine the way I am doing it. Here $\epsilon$ change the first entry on string ...
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19 views

What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$?

What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$? (the normal core of $H$ in $G$ is the largest subgroup of $H$ which is normal in $G$. it is the intersection of all ...
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52 views

Problems with a proof involving graphs and groups

I'm studying an article that is the main literature when it comes to non-commuting graph : this article. Originally, a non-commuting graph of a group (denoted by $\Gamma_G$) is a graph whose vertices ...
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4answers
58 views

Prove the only homomorphism between groups with coprime orders is trivial. [closed]

Deduce that if $\gcd(|G|, |H|)=1$ then the only homomorphism $\phi : G \rightarrow H$ is trivial.
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1answer
30 views

Q has no maximal subgroups.

Theorem: If $R$ is a ring with 1 and $I$ is a ideal in $R$ such that $I \neq R$, then there is maximal ideal $M$ of the same kind as $I$ such that $I\subseteq M$. Note:- IF $R$ has no unity it is not ...
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13 views

Prove the order of the group homomorphism of an element divides the order of the element.

Let $\phi : G \rightarrow H$ be a group homomorphism. Prove $\forall g \in G$, the order of $\phi(g)$ divides $g$. I've gotten to the point where I've shown that if, $ord(\phi(g)) < ord(g)$ then ...
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17 views

Decomposition of an algebraic group into unipotent radical, component group and Levi subgroup

Consider the following statement : Given an algebraic group $G$ (over $\mathbb{R}$ or $\mathbb{C}$), there is a decomposition $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
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18 views

Problem showing Newtons Laws are invariant under the Euclidean Group

I am trying to show that the equations of motions of physics are invariant under the Euclidean group $E_N$ for $N=3$. Therefore we have Newton's Laws as: $$m\frac{d^2 ...
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34 views

Isomorphisms with factor groups

Let H and K be normal subgroups such that H$\vee$K=G and H$\cap{K}$=$\left\{{e}\right\}$, where H$\vee$K=. Prove that G/H$\simeq$K and that G/K$\simeq$H. I know elements of H$\vee$K should be of the ...
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39 views

Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
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31 views

Classifying groups of order 4

Let $G$ be a group of order $4$ then either $ G \cong C_4$ or $G \cong C_2 \times C_2$ the proof my lecture gave goes as follows: Let $x \in G$ then by Lagrange $\text{ord}(x)$ divides $|G| =4 $ so ...
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22 views

Prove Grigorchuk group is self similar.

Where I can find a proof of self similarity of Grigorchuk group. I read it somewhere that Grig group follows this interesting property so I read about it but could not find a proof anywhere. It was ...
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54 views

What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements?

My formatting didn't work in the title, here is the question again: What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements? I know that this group does not have order 24 ...
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36 views

Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
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25 views

Examples for Burnside problem.

What are some examples for Burnside Problem- example of an infinite finitely generated torsion group - except Grigorchuk group. I have studies Grigorchuk group as an counterexample which was first ...
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1answer
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Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
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1answer
32 views

About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
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1answer
50 views

Is GRP a subcategory of SET, or not? [duplicate]

This is the notion of a subcategory $\mathscr{D}$ of a given category $\mathscr{C}$ which I use: it consists of a subcollection of the collection of objects of $\mathscr{C}$ and a subcollection of the ...
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1answer
16 views

Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
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1answer
38 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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1answer
43 views

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$?

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$
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2answers
77 views

Are $(\mathbb{R} - 0, \times)$ and $(\mathbb{R}, +)$ the same group? What is its name?

I'm trying to justify to a friend why it's "not a coincidence" that $a^ba^c = a^{b+c}$, and I want to argue that it's because the structure of $\mathbb{R}$ under addition is exactly the same as the ...
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1answer
51 views

If $\left<3\right> \cong \mathbb Z_2$ and $\left<2\right> \cong \mathbb Z_3$, why isn't $\left<3\right>\left<2\right> \cong \mathbb Z_2\mathbb Z_3$

$\left|\mathbb Z_{6}\right|=6=2 \cdot 3$. Since $\mathbb Z_{6}$ is abelian, all subgroups are normal and thus its Sylow subgroups are unique. $\left<3\right>$ is the $2$-Sylow, and ...
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53 views

Annihilating Ideal of a Ring

I am stuck on how to show this. A starting hint would be helpful, and an answer (hidden) would be much appreciated. I tried supposing that there was another element in the annihilating ideal, however, ...
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When is a group isomorphic to a proper subgroup of itself?

A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?
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Why isn't $\mathbb Z_9 \cong \mathbb Z_3 \times \mathbb Z_3$ by the fundamental theorem of finite abelian groups?

I was reading the answer to this question: Explicit descriptions of groups of order 45 and the accepted answer says the Sylow $3$-subgroup is either isomorphic to $\mathbb Z_9$ or $\mathbb Z_3 \times ...
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Group $G$ acts on set $Ω$. $|G| = 30$, $Ω$ has size 3.

Five elements of $G$ fix every element of $Ω$. What subgroup sizes are guaranteed? Without the fixed elements, by Sylow E and Cauchy theorems, there should be $1,2,3,5,6,10,15$. Are all of the ...
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23 views

Proving relations between kernels and images of a Group G

Let $G$ be an abelian group and n be an integer. Define the map $\phi_n\colon G \to G$, $\phi_n(g) = g^n$ since $G$ is abelian $(hg)^n = h^ng^n$ that is $\phi_n$ is a homomorphism. We then have the ...
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$|G| = 1155$, $N \lhd G$, $|N| = 55$, $K \leq G$, $|K|=35$. $|<N,K>|$ and $|N \cap K|$?

since $gcd(|G:K|,|N|) \neq 1$, I can't use $NK=K$ and $N \cap K = N$. I tried using Sylow $p$-subgroups, but they don't seem to help this problem. Does $NK \lhd G$ have to be true? Also are ...
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1answer
55 views

Various Intersections of Sylow p-subgroups.

I was told yesterday that in a system of Sylow $p$-subgroups of a finite group $G$, if, $\{S_1,S_2, \cdots, S_n\}$ make up the system, it can happen that, say, the intersection of $S_1$ and $S_2$ has ...
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1answer
20 views

Consider $\phi: G \to S_4\times D_{15}$ a homomorphism and onto

Q1) Prove that $G$ contains an element of order 20 Q2) Assume $\exists H\subset G$ s.t $H$ normal in $G$ and |$\phi(H)$|=60. Prove that $G$ contains a normal subgroup $K$ such that |$G/K$|=36 For ...
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2answers
104 views

Algorithm design for enumerating pairs of noncommuting elements up to conjugacy

I am trying to write some Magma code that, given a group $G$, returns a list of pairs $(x,y)$ with $x,y\in G$ such that $[x,y]\neq 1$ and such that every pair $(z,w)$ in the group with $[z,w]\neq 1$ ...
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1answer
23 views

Free product of infinite many groups is not finitely generated?

Let $\{G_i\mid i\in I\}$ an infinite family of (not trivial) groups. Is it true that the free product $\ast_{i\in I} G_i$ is not finitely generated? I think it's true, I just need confirmation.
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66 views

Elements of order three in $GL_3(2)$

How do I go about finding elements of order 3 in $GL_3(2)$? I'm currently trying to show that the automorphism group of a Klein 4-group induced by conjugation in $GL_3(2)$ is isomorphic to $S_3$ so am ...