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

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Find a group $G$ which contains the elements $a,b,c$ such that $a\ne b$ and $ac=cb$

The title says it all. I'm trying to find a group $G$ which contains the elements $a,b,c$ such that $a\ne b$ and $ac=cb$. I didn't have an idea how to construct the group $G$ in a smart way so I was ...
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1answer
13 views

Generators of $Sp(2n)$

Let $\sigma =\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$. Define $J_{2n} = \underbrace{\sigma \oplus \cdots \oplus \sigma}_{\text{$n$ copy}}$. We define a $2n \times 2n$ real matrix matrix ...
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0answers
40 views

How the centers of a group and its subgroup are related?

Let $G$ be a group and $H$ be a subgroup of $G$. What can I say about the centers of $G$ and $H$. How are they related? For example, if I know the center of $GL(n,\mathbb{R})$ is ...
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16 views

How many permutations of a linear equation

How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.) I believe the ...
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1answer
25 views

Are Sylow p-subgroups in conjugate?

Let $G$ be an infinite group and $p$ be a prime number. Let $\mathscr{C}$ be a chain of p-subgroups of $G$ ordered by inclusion. Then, for every element of the union of $\mathscr{C}$ has an order ...
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2answers
35 views

Prove that $|A\cap B| \le \frac {1}{2} |A|$ where $A,B$ are two subgroups of $G$

Suppose $G$ is a finite group, $A,B$ are subgroups of $G$ and $A$ isn't a subgroup of $B$. Prove (by using Lagrange's theorem) that $|A\cap B| \le \frac {1}{2} |A|$. $ $ This is what I have so far: ...
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3answers
33 views

Infinite group with finite order elements

Can you give me an example of an infinite group in which every element has order $3$ (except identity) ?
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51 views

Exercise 1: Galois Theory (J. Rotman)

Definition: Let $F$ a figure in the plane, its symmetry group is defined by $\Sigma(F):=\{\sigma \in O(2,\Bbb R)\mid \sigma(F)=F\}$. Here $O(2,\Bbb R)$ denotes the real orthogonal group. Exercise 1: ...
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54 views

order of non abelian group can't be what?

Let $G$ be a non abelian group; then its order can be: $25$ $55$ $35$ $125$ I think the order cannot be $25$ and $35$. But from option $55$ and $125$ which one is not possible? Why not $25$ ...
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1answer
27 views

Questions in Abstract Algebra

I have two question which I couldn't solve: Let $G$ be a group of size $40$. a. Show the $5$-Sylow subgroup in $G$ is Normal - this part was easy, I just showed that $n5=1$ and then $P5$ is ...
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27 views

$\langle s \rangle \sim \langle t \rangle$ and $\langle s \rangle \cap \langle t \rangle$, where $s$ and $t$ are permutations in $S_6$

Let $s=(12)(345)$ and $t=(123456)$ be permutations in $S_6$ how to know if $\langle s \rangle$ and $\langle t \rangle$ are isomorphic or not? Also what about $\langle s \rangle \cap \langle t ...
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1answer
20 views

Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
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1answer
22 views

Primitive Roots Modulo $2^n$ for $n\geq3$

Question: (a) Prove that there is no primitive root modulo $2^n$ for any $n\geq3$, where $\bar{a}\in(\mathbb{Z}/2^n\mathbb{Z})^\ast$ is a primitive root modulo $2^n$ if the order of $\bar{a}$ is ...
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0answers
18 views

Specific Subgroups of an Abelian Group

I am looking for an elementary proof of the following result: If G is a finite abelian group and H is a subgroup of G, then G contains a subgroup isomorphic to G/H. This can be proved rather easily ...
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0answers
32 views

A Problem from I Martin Isaacs Algebra: A graduate course [duplicate]

Suppose G = H U K U L where H,K, and L are proper subgroups of G. Prove that [G:H]=[G:K]=[G:L]=2. I have trouble believing this, let alone proving it. Thanks for any pointers.... or a proof! Gary
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1answer
30 views

Do there exist pro-$p$ groups with finite quotients of non $p$ power order?

We define a pro-$p$ group to be a projective (i.e. inverse) limit of $p$-groups. My question is exactly as stated in the title: If a subgroup $H$ of a pro-$p$ group $G$ has finite quotient, must ...
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1answer
22 views

Inequivalent representations of a finite group

I'm looking for this result: A finite group has only finitely many inequivalent representations of given degree over a field of characteristic $0$. Do someone know where I can find a proof of ...
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0answers
19 views

Generators of $\Gamma_0(N)$

Let $\textbf{T}:=\bigl(\begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \bigr)$, $\textbf{S}:=\bigl(\begin{smallmatrix} 0&1/\sqrt{N}\\ -\sqrt{N}&0 \end{smallmatrix} \bigr)$ and $H$ the ...
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1answer
42 views

Homology group $H_1(G;\mathbb{R})$ is a vector space?

I am reading a paper which is asking me to view the homology group $H_1(G;\mathbb{R})$ of a (presentation of a) group as a vector space. Now, my knowledge of homology is basically non-existent, but I ...
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1answer
25 views

A normal intermediate subgroup in L30 lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in L30 lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups ...
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11 views

A relationship between central-by-finite groups and FC-groups II [on hold]

Just now asked a question with the same title. Now I would like to improve my question with the following. Let $G$ be a locally finite group. Suppose that $G$ is a FC-group. Let $x$ be a element of ...
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1answer
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A relationship between central-by-finite groups and FC-groups

A group is said FC-group if for all $x\in G$ is true that the set $x^G$ is finite. Equivalently, $G$ is a FC-group if $|G:C_G(x)|$ is finite for all $x \in G$. A group is said a central-by-finite if ...
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14 views

Group theory and centres [duplicate]

If $G$ is a $p$-group and $H$ is a non trivial normal subgroup of $G$, how do I show that the size of $H\cap Z(G)$ (where $Z(G)$ is the centre of $G$) is $\ge p$? A hint is given to consider the ...
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40 views

True False Statements (Group Theory )

Let $G_1$ be an Abelian group of order $6$ and $G_2$ = $S_3$. a. Both $G_1$ and $G_2$ have unique subgroup of order $2$. b. Neither $G_1$ and $G_2$ have unique subgroup of order $2$. c. $G_1$ ...
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31 views

Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...
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0answers
19 views

Order of subgroup image in relation to original groups

There is a homomorphism between two groups G1 and G2, and H1 is a subgroup of G1. If H2 is defined as the image of H1, then I need to show that if H2 is finite, |H2| divides |G2|, and that if G1 is ...
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0answers
11 views

Cyclic group generator and Multicative identity iof Correspondng Ring

Can cyclic groups made into ring with unity such that multiplicative identity is not any generator?(Or does there exist example of one such cyclic group) Can we make (Z, +) into ring with unity ...
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1answer
27 views

The order of its elements in the additive group $\mathbb{Z}/9\mathbb{Z}$.

I want to determine the order of each element in the additive group $\mathbb{Z}/9\mathbb{Z}=\lbrace \bar{0},\bar{1},\bar{2},\dots, \bar{8}\rbrace$. It is taken from the Example (4) page 20 in Abstract ...
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3answers
471 views

Why is Multiplicative Notation Used for Groups (Instead of Additive)?

In documents relating to group theory it seems common to use a multiplicative notation to represent the group operation. For example, I'm reading Herstein's "Topics in Algebra" and looking for some ...
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1answer
31 views

what is the maximom order of an element is $\mathbb S_{15}$ [duplicate]

Let $S_n$ be the permutation group of order $n$. What is the maximom order of an element on $\mathbb S_{15}$. Is there any way to find maximom order of an element in $\mathbb S_{15}$ or find the ...
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16 views

How does the $10$ dimensional irrep (tensor) of $SU(3)$ look like?

We know that for $SU(3)$ the following tensors furnish the $\mathbf{d}$ dimensional irreducible representation: $$\phi^i\hspace{1cm} (\mathbf{3})\\ \phi^{ij}\hspace{1cm} (\text{asymmetric in ...
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2answers
40 views

If $G = \{\sigma \in S_n\mid \epsilon(\sigma) = 1\}$ then is $(G,\ast)$ a group?

Let $G = \{\sigma \in S_n\mid \epsilon(\sigma) = 1\}$ with some $n\in\mathbb{N}$, $\ast$ is the multiplication (i.e. composition) of permutations. Is $(G,\ast)$ a group? I have some questions ...
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83 views

Inverse limit of $\mathbb{Z}/n\mathbb{Z}$

I know that this is well-known fact that $$\lim\limits_\leftarrow\mathbb{Z}/n\mathbb{Z}=\prod\limits_p\mathbb{Z}_p,$$ however I don't know the rigorous proof of this. Can anyone give me the ...
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1answer
37 views

I need to identify $G/H$ upto isomorphism. [duplicate]

$\varphi:G=(\mathbb{C}^*,.)\to (\mathbb{C}^*,.)$ given by $\varphi(z)=z^4$ clearly Ker$\varphi=H=\{z:z^4=1\}=\{1,-1,i,-i\}$ I need to identify $G/H$ upto isomorphism. Thanks for helping.
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1answer
24 views

Check whether subgroup is normal in $ S_4$

Let H={e,(12)(34)} and K ={e, (12)(34), (13)(24), (14)(23)} be subgroups of $ S_4$ where e is identity element. Then which of following is true H and K are normal subgroups of $S_4$ H is normal in K ...
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1answer
30 views

Classifying groups of order $2p^2$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text (p. 185 Exercise 15): Let $p$ be an odd prime. Prove that every element of order $2$ in $GL_2(\mathbb{F}_p)$ ...
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1answer
62 views

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] ...
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2answers
51 views

$(\mathcal{P}(\mathbb{N}),\cap)$ - is a group or not?

I am trying to prove that $(\mathcal{P}(\mathbb{N}),\cap)$ is a group but I'm not really sure my proof is correct. When checking for the identity element I found that for every $A$ in ...
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1answer
63 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
24 views

Does $G\to X$ have dense image iff $T_eG \to T_{\theta(e)}X$ is surjective?

Let $G$ be a connected algebraic group, $X$ a variety and and $\theta : G\to X$ be a morphism of varieties. (In particular it could be the orbit of some action of $G$.) Consider the corresponding ...
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Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
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Good reference for The Differntiable Slice Theorem

I am looking for a book that will give me a good proof of The Differentiable Slice Theorem - Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Then every orbit has a $G$-invarient ...
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1answer
24 views

Order of normalizer in $S_6$

Find order of normalizer of permutation $s= (12)(34) \in S_6.$ I tried it and I thought we need all permutations $p$ s.t $psp^{-1}=s,$ I wrote down such $p,$ I counted $8.$ But in book answer is ...
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1answer
17 views

Prove that $P_1\cap P_2$ is $p$-Sylow subgroup of $N_{G}\left(P_1\right)\cap N_{G}\left(P_2\right)$

Let $P_1, P_2$ be $p$-Sylow subgroups of $G$, Show that $P_1\cap P_2$ is $p$-Sylow subgroup of $N_{G}\left(P_1\right)\cap N_{G}\left(P_2\right)$. Don't have any idea.. Thanks !
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1answer
48 views

A normal intermediate subgroup in L30 lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$. An intermediate subgroup $H \subset K \subset G$ is a ...
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2answers
38 views

Find $o\left (\frac{G}{Z (G)}\right) $ [on hold]

Let $G :=\{a^k, a^k.b|0\le k\lt 9\} $ s..t $o(a)=9$ and $o(b)=2$ and $ba= a^{-1}b.$ If $Z(G)$ denotes center of group $G,$ find the order of $G/Z(G).$ In book answer is $18.$
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For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

A group $G$ and a function $f:G\longrightarrow G$ are given and for all $x,y\in G$ satisfying $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?
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21 views

Show that $G$ contains elements $a,b$ s.t. $a^2=b^3=e$ and $ aba=b^2=b^{-1}$

Given that $|G|=6$ and is not commutative. I've showed that generators of $G$ have periods either $2$ or $3$, since $|H_a|=\text{period of generator a}$ divides $|G|$. Since $G$ is not commutative it ...
2
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0answers
32 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
3
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1answer
41 views

When a normal subgroup $N$ admits a complament?

Let $G$ be a finite group and let $N$ be a normal subgroup. I am looking for conditions on $N$ (and maybe also on $G$) such that there exist a subgroup $H$ of $G$ such that $$G=N\rtimes H.$$ Clearly, ...