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

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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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34 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
24 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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1answer
24 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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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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16 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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30 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
29 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
20 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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18 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
36 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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17 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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10 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
13 views

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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39 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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28 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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18 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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9 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
25 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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467 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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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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38 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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76 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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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
59 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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$(\mathcal{P}(\mathbb{N}),\cap)$ - 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
60 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
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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
44 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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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 ...
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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 ...
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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, ...
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Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
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1answer
44 views

Find orbit of $1$ for $\sigma$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ $ 1 \mathop{\rightarrow}^{\sigma} 3 \mathop{\rightarrow}^{\sigma} 4 ...
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3answers
110 views

When are two direct products of groups isomorphic?

I was thinking about the following problem: Suppose that $G_1 \cong G_2$ are isomorphic groups. Under what conditions on the groups $H_1,H_2$ will we have $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
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1answer
38 views

Prove that the group algebras $\mathbb{C}\mathbb{Q}_8$ and $\mathbb{C}\mathbb{D}_4$ are isomorphic. [on hold]

I need to prove that group algebras $\mathbb{C}\mathbb{Q}_8$ and $\mathbb{C}\mathbb{D}_4$ are isomorphic. How can i do this?
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45 views

Let $G$ be a group and $u \in G$ be a fixed element. By the following, prove that $(G,\bullet)$ is a group.

Let $G$ be a group and $u \in G$ be a fixed element. Define the operation $\bullet$ on G as $\forall a,b \in G, a \bullet b=au^{-1}b.$ Prove that $(G,\bullet)$ is a group. So, I know that in order ...
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20 views

Is $BU^-$ open in GL(n,C)?

Given $G= GL(n, \mathbb{C})$ seen as a Lie Group, let B be the Borel subgroup of upper triangular matrices and $U^-$ be the subgroup of unipotent lower triangular matrices (i.e. lower triangular ...
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7 views

Rewriting a sum of Young Tableaux as Tensors

Is there a straightforward way (perhaps a software) that can write a direct sum of Young Tableaux in terms of tensors? For instance the direct product in $SU(3)$ (taken from this post) ...