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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$\Gamma(2)$ and $\Gamma_0(4)$ are conjugate

Show that the groups $\Gamma(2)$ and $\Gamma_0(4)$ are conjugate to each other in $SL(2,\mathbb{R})$, where $\Gamma(2)$ and $\Gamma_0(4)$ is congruence subgroups of the modular group Is it the ...
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A probabilistic algorithm (something close to Discrete logarithm)

$p$ - is a prime number. $a\in Z_p^*$ is a creator of the group $Z_p ^*$. The definition of the Discrete logarithm of $c\in Z_p^*$ is: $$\log_a c=b\iff a^b=c \mod p $$ Assume we have an algorithm ...
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Prove that $x * y = \frac{x+y}{1+xy}$ is a stable part of $G=(-1, 1)$

I have to prove that the result of $x * y \in G$ so $\frac{x+y}{1+xy} \in (-1, 1)$. So $x > -1$ and $y > -1$ at the same time $x < 1$ and $y < 1$. If I multiply the first 2 expressions I ...
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31 views

Relaxing the subgroup requirement for cosets

The definition I have for left cosets is as follows: Let $G$ be a group and let $H$ be a subgroup of $G$. A left coset of $H$ in $G$ is a set of the form $gH=\{gh:h\in H\}$ for some $g\in G$. ...
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Induced homomorphism $\phi^*$ from $G/M \to H/N\ ?$

Let $\phi : G \to H/N$ be a homomorphism where $G$ and $H$ are groups and let $M \unlhd G$ and $N \unlhd H$. Now when does $\phi$ induces a homomorphism $\phi^*$ from $G/M \to H/N\ ?$ When $M ...
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66 views

A few tasks from group theory

Can you tell me, if my solutions are good? Mapping $f:\mathbb{Z}\rightarrow G$ with $f(k)=g^k$ is group homomorphism and Image(f) is abelian subgroup of G with $|\langle g \rangle |=ord(g)$ ...
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23 views

cyclic subgroups and generators

g is a generator of the cyclic group C45. Sketch the lattice of subgroups of $C_{45} $ For each subgroup, state its order and give a generator in terms of g. You do NOT need to list the ...
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20 views

Define all points in the affine integral lattice.

Define all points in the affine integral lattice $\mathcal{L}=\{(x,y,z,t) : x+y+z+t=5$ and $x-z \equiv 0$ (mod $12$)$\} \subset \mathbb{Z}^4$. This is a question from a practice exam I have with no ...
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25 views

Group homomorphism identify kernel

Show that the following map is a group homomorphism and find its kernel. State whether the mapping is injective or surjective. $\phi : \mathbb{Z} \to (\{1,-1\},{\times})$ by $\phi(a) = ...
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31 views

Exploiting geometric invariants via group theory

Let $T$ be the set of all plane triangles. The problem is to find $t \in T$ s.t.h. a predicate $P(t)$ holds. At present, I'm doing this by a form of randomized search procedure (effectively via a ...
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43 views

How to obtain real irreducible representation matrices for finite point groups?

I would like to generate the irreducible representation matrices in real (not complex) form for any finite point group, in order to use them in a projection operator. At least I require the diagonal ...
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27 views

Show that the order of G = order f(G) times order ker(G).

Let $f:G \rightarrow G'$ be a homomorphism and let $H$ be the kernel of $G$. Suppose $G$ is finite. Show ord$(G)=$ord$(f(G)) \cdot $ord$(H)$. What I want to do is to construct a bijection, $\Phi$ ...
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1answer
24 views

Is there a general formula for finding all subgroups of dihedral groups?

It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},...,\{e,r^{n-1}s\}, \{e,r,r^2,...,r^{n-1}\}, D_n$ are always subgroups of $D_n$. Especially when $n$ is odd, these seem to be the only subgroups. ...
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1answer
32 views

A group of even order must contain an odd number of elements of order 2. [duplicate]

I tried it using proof by contradiction: Suppose that there are even number of elements of order $2$. Call them $x_1, ..., x_{2m}$, where $x_1,...,x_{2m} \neq e$. Then, consider the set $G=G ...
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Don't quite understand the question. [Rotman: Group Theory ][Semantics question]

This is more of a "If I knew what that symbol/word/phrase meant I might understand the whole Idea" question and less of a "I am so lost I cannot even understand the question" question. Here is the ...
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Why does $\{1 \dots 9\}$ behave like this under multiplication mod $10$?

When I multiply the set $$\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$$ by $2$ and take the remainder mod $10$, I get the following repeated pattern. $$\{2, 4, 6, 8, 0, 2, 4, 6, 8\}$$ Multiplication by any even ...
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The multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$

Let $G$ be the multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$. Then assess the following claims: Every proper subgroup of $G$ is finite. $G$ ...
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24 views

List of two-sided wallpaper groups?

I'm interested in the symmetries of two-dimensional patterns that have two sides. In other words, what discrete groups can be formed from the three-dimensional Euclidean isometries which preserve a ...
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38 views

Question about the assumptions to have $G \simeq H\times K$

I've been looking this fact: Let $G$ be a group, with $G$ abelian. Let $H$, $K \leq G$, with $G=HK$ and $H\cap K=\{e\}$. Then, we have that $G \simeq H\times K$. And my question is: We know ...
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Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
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29 views

Matrices within Group Theory

Recall that $GL_{2}(\mathbb{R})$ denotes the group of 2x2 invertible matrices with real entries with the product given by matrix multiplication. Let H denote the smallest subgroup of ...
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35 views

Orbit, Stabiliser help please?

Let $Q$ denote the rectangle with the vertices $C_1=(2,1), C_2=(-2,1), C_3=(-2,-1)$ and $C_4=(2,-1)$. Describe the elements of the symmetry group $G$, of $Q$. Note that $G$ permutes the edges of ...
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Find two finite abelian groups of same order $G$ e $H$ such that $G \ncong H.$

Find two finite abelian groups of same order $G$ e $H$ such that $G \ncong H.$ I found groups $\mathbb{Z}_4$ e $V$ (Klein's group) that satisfy it, but would like more examples. I'm trying to use the ...
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Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
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Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
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28 views

Why is $\operatorname{Hom}_{\mathrm{Groups}}(G,A)$ isomorphic to $\operatorname{Hom}_{\mathrm{Ab}}(G/[G,G],A)$?

This question is inspired by an exercise from the Weibel's book on Homological Algebra (beginning of chapter 6 on Group Cohomology). Let $G$ be a group and $A$ be a $G$-module. My question simply is: ...
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Looking for group of polynomials with only real roots

Assume $P_\mathbb R$ is the set of all polynomials which have only real coefficients and only real roots. Define $0$ as a polynomial with infinitely many real roots and all other constant polynomials ...
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Induction of an irreducible group representation

I'm having some trouble finding the answer to the following question. Any ideas on how to get started? Let $H$ be a subgroup of a group $G$ and let $U_{1}$, ...,$U_{k}$ be the irreducible ...
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$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
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Help with translating from French to English

In a paper by G. Boccara, "Cycles comme produit de deux permutations de classes données," I have come across something that seems weird. On page 130, notation 1.7, it says: Si $l$ est un entier ...
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2-Frobenius groups of order $2^{10}.3^5.5.11$

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
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Partial Group with elements only comparable to 0?

The use case is for managing a digital ledger where the monetary amounts are kept private, but transactions are still verifiable through addition. To do this, I need a mathematical construct which ...
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(Non-Hopfian) groups that only have quotients that are themselves or the trivial group.

A group is non-Hopfian provided it is isomorphic to a proper quotient. The classic, finitely presented, example of such a group is the Baumslag-Solitar group $$BS(2,3)= \langle x,t \mid t^{-1}x^2 t ...
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If $C_G(H)=N_G(H)$ for all abelian subgroups, prove that $G$ is abelian

Let $G$ be a finite group such that for all abelian subgroups $H$ of $G$, $$C_G(H)=N_G(H).$$ Prove that $G$ is abelian. ($C_G(H)$ is the centralizer, $N_G(H)$ is the normalizer of $H$ in $G$) my ...
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Does group $G$ of order 42 have a normal cyclic subgroup of order 21?

Show that a group $G$ of order 42 has a normal cyclic subgroup of order 21. What I did so far is using Sylow's theorem to show that $G$ has a unique 7-sylow subgroup $S(7)$ (which is normal) and {1 ...
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Count number of colorings of tetrahedron, where colorings are indistinguishible if one can be reached from another by rotation

I'm fairly new to group theory, and here's one problem I'm trying to solve: We're coloring nodes of tetrahedron in 3 distinct colors, and its edges in 2 distinct colors. We're treating two colorings ...
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What does $SL_2(\mathbb{F}_q)/\{I,-I\}$ look like?

I am not sure what the elements of $$SL_2(\mathbb{F}_q)\big/\left\{ \begin{pmatrix}1&0\\0&1\end{pmatrix},\begin{pmatrix}-1&0\\0&-1\end{pmatrix}\right\}$$ look like for an arbitrary ...
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Stabilisers, Orbits and Group Theory Help?

a) Draw a regular pentagon with vertices ${v_{1} ,...,v_{5}} \subset \mathbf{R}^2$ such that $v_{1}$ has coordinates (1,0) and the origin in the centre of the pentagon. For each reflection symmetry ...
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Is this a better way to think about Groups as Categories?

I asked a bit ago how to reconcile the category theoretic and set theoretic definitions of groups (groupoid which is a monoid vs the set theoretic definition), and I got the answer I was looking ...
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Prove Zs, Gs (the group of symmetries of the square) and the quaternion group Q are not pairwise isomorphic.

Prove $Zs, Gs$ (the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic. How would you go about proving. Seems quite difficult. I know that none of the latter ...
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Finding the order of the set of elements fixed by all elements of a group.

I've got an old exam question that I can't figure out how to solve. If anyone could let me know what theorems and lemma's I might find useful, please let me know. Let G be a group of order $p^m$ for ...
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Is every normal subgroup of a finitely generated free group a normal closure of a finite set?

Let $G$ be a group, $S\subset G$ a subset, then the smallest normal subgroup of $G$ which contains $S$ is called the normal closure of $S$, and denoted by $S^G$. My question is, if $G$ is a free ...
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A peculiar decomposition of elements in a group

Let $G$ be an abelian group. Suppose there exists $n\in \mathbb N$ such that $\forall x\in G, x^n=e$. Let $a,b \in \mathbb N$ such $ab=n$ and $\gcd(a,b)=1$. Let $G_a=\{x^a \; | x\in G\}$ and ...
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Uniqueness of a subgroup of a given order

Let $G$ be a cyclic group with order $n$. Prove that for every divisor $d$ of $n$ there is a unique subgroup with order $d$. For the existence, let $x$ be a generator of $G$. It is easy to check ...
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Finding conjugacy classes

I've been having problems with finding conjugacy classes. I don't really understand how to do it properly. Say we look at a S3 group: $S_3=]e, (12), (13), (23), (123), (132)]$ If we look at just ...
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Is this enough to prove that the group is isomorphic to $S_3$?

I have a relatively complicated group, I will not go into detail about what it is, it a group of automorphisms, and the group-relation is composition, so it is kind of complicated. However, I am ...
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If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.

If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$. I want to use induction to prove this: It is trivial when ...
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Size of conjugacy classes of alternating group $A_{22}$

Let $p,q\in\{13,17,19\}$ and $G=A_{22}$, is it true that for every $x\in G$ we have $(|x^G|,pq)\neq 1$? why?
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On finite groups with a special property on proper subgroups [closed]

Let $G$ be a finite group such that all proper subgroups of $G$ are nilpotent. Then $G$ is solvable.
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How to create lattice diagrame in maple 14?

I am studying lattice diagrame of subgroups of groups and I have already posted one query over here. Now my present query is: I am using MAPLE 14. Can anyone suggest me how to create lattice ...