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

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Presentation of a group isomorphic to A_4

I have a group G defined by $G = <x,y,z|x^2 = y^3 = z^3 = xyz>$ and we know that $a$ $=$ $xyz$ belongs to the centre of G. But im struggling to show that $\frac{G}{<a>} \cong A_4$. Seeing ...
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7 views

Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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1answer
9 views

Power of two commuting elements in a group is the binary operation of each of the two elements raised to that power

Let $(G,\ast)$ be a group and let $n\in\aleph$. Prove that if g, h $\in G$ commute, then $(g\ast h)^n$=$g^n\ast h^n$
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0answers
13 views

How to expand powers of multiple pairwise commuting elements in a group [on hold]

Let (G, $\ast$) be a group and let n $\in\aleph$. Prove that if $g_1,...,g_k\in G, k\in\aleph$ are pairwise commuting elements of G, then $(g_1\ast...\ast g_k)^n$=$g_1^n\ast ...\ast g_k^n$
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21 views

The center and centralizer of a group.

Show that a∈Z(G) if and only if C(a)=G. Is this a proof by contradiction break
3
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1answer
42 views

Can every group be extended to ring with idenity [duplicate]

Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But ...
1
vote
1answer
41 views

Every finite Set as non-abelian Group

For what values of n, we can find a non abelian group. The facts I have proved till now: 1. For n prime there exist only one group upto isomorphism which is cyclic hence abelian 2. For n = 4, there ...
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0answers
42 views

Simplifying a direct sum $\mathbf{3}\oplus\mathbf{3}\oplus\mathbf{2}$ etc

In particle physics, one often uses the dimensionality of the irrep to label the irrep (apparently this is not a very good idea since the dimension does not unambiguously determine the rep.). What are ...
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20 views

Does the decomposition of a Lie group manifold imply a type of group product?

The real symplectic group manifold is diffeomorphic to this Cartesian product of manifolds: \begin{equation} \operatorname{Sp}(2n,\mathbb{R}) \simeq \operatorname{U}(n) \times \mathbb{R}^{n(n+1)}. ...
2
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2answers
55 views

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$?

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$ as $\mathbb{Z}$-module over $\mathbb{Z}$? My proof: Define surjective function $f:3\mathbb{Z}/15\mathbb{Z} \rightarrow ...
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33 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
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0answers
38 views

Under which conditions two groups of order $n=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ are isomorphic

$G,H$ two groups of order $n=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ under which of the following conditions $G$ isomorphic to $H$ (prove or give a counterexample) 1) $G,H$ have have same ...
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0answers
27 views

Prove that all groups of order $3^k5^l$ solvable given $k \le 3$

Prove that $G$ is solvable given its order is $3^k5^l$ while $k,l \in \mathbb{N} , k \le 3$. we are not allowed to use burnside's theorem and Feit–Thompson. I tried to use sylow's theorems to prove ...
1
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1answer
29 views

How many possible isomorphisms do we have between G and H? [duplicate]

Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order ...
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0answers
26 views

Sylow subgroups of Symmetric Group

The symmetric group (=permutation group) $S_n$ acts on the set $X_n$ of polynomials in $n$ variables $x_1, x_2, \cdots, x_n$ [with coefficients from $\mathbb{Z}/ \mathbb{Q}/$ or any ring of ...
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1answer
23 views

Simple homomorphism of groups question [duplicate]

Show that a homomorphism of groups also has the property that $f(a^{-1})=f(a)^{-1}$ for all $a \in G$.
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1answer
25 views

find a special group

Could anyone show me a virtually-nilpotent ( finitely generated, countable discrete) group $G$ such that $G$ is neither finite-by-nilpotent nor virtually abelian? Thanks! Remarks: 1, By a result ...
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0answers
16 views

Find up to isomorphism all the quotient groups of composition series of a group of order $30$.

I can't seem to understand what I should do here... All I did so far is proving that $G$, (such a group), is not simple. But there are many cases, I can't really tell what $n_2,n_3$ and $n_5$ are, ...
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1answer
17 views

Inverse of a product in a group can be wriiten as the product of the inverses of each element in reverse order

Let $(G,\circ)$ be a group and let $g_1,...,g_n\in G, n\in\aleph$. Prove that $(g_1\circ ...\circ g_n)^{-1}=g_n^{-1}\circ ...\circ g_1^{-1}$ I tried this by induction but was unsure how to take out ...
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2answers
24 views

Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
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1answer
15 views

If G is a compact semisimple Lie group and Z is its center, is G/Z always compact?

The title pretty much sums up the question: Suppose $G$ is a compact semisimple Lie group with center $Z$, The question is if $G/Z$ is always compact? or, under which conditions will it be compact?
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45 views

Commutative generators of a group

If a group has commutative generators is the group always abelian? I have a question dealing with how to determine if a Cayley graph of a group is an abelian group. It seems that if the generators ...
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2answers
50 views

How many homomorphisms are there from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$?

I need to determine how many homomorphisms there are from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$. I have never solved that kind of question. I do know that orders are preserved and that some elements can be ...
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0answers
37 views

Degrees of irreducible complex characters of alternating groups

What is sum of degrees of the irreducible complex characters of the alternating groups? The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra ...
2
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1answer
49 views

Group of order 396 isn't simple

Prove that group of order $396=11\cdot2^2\cdot3^2$ is not simple. $n_{11}$ is $1$ or $12$, so I assumed $n_{11}=12$ and tried to look at the action of the group on $Syl_{11}\left(G\right)$ by ...
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3answers
33 views

Question about group theory and order of elements

Let $G$ be a group and $x, y \in G$. Prove that $ord(x)=ord(y^{-1}xy).$ Let $n,m$ be integers such as $x^n=1$ and $(y^{-1}xy)^m=1$. $x^n=(y^{-1}xy)^m=y^{-1}x^my=1$ I'm not sure how should I ...
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49 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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24 views

Bigenetic properties of finite group [on hold]

Nilpotency, supersolubility and polycyclicity are bigenetic properties of the class of all finite group. Let be : P is property, X be a class o group. We say that P is a bigenetic property of ...
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27 views

orbits from group [on hold]

Let g is the group Z2 and let 56 points as follows: w:= [ 7, 8, 15, 27, 42, 89, 95, 121, 125, 134, 139, 150, 167, 5 , 10, 11, 18, 30, 45, 92, 98, 124, 128, 137, 142, 153, 170, 8 , 12, 13, 20, ...
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1answer
49 views

Number of units of $\mathbb{Z}/11\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$

Let $n\mathbb{Z} = \{nk\::\:k \in \mathbb{Z} \}$. Find the number of units of $\mathbb{Z}/11\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$. I tried this problem by using the fact that since ...
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23 views

What is the root structure of the Diffeomorphism Group?

Being a physicist, I think it'd be cool to have Coxeter plane projections of the root systems of the symmetry groups associated with the fundamental forces hanging on my walls (example for E8: ...
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1answer
32 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$? [on hold]

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
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2answers
77 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
2
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1answer
26 views

$V^{\oplus3}$, linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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2answers
30 views

all the squares in the multiplicative group $\mathbb{Z}_n^*$. [on hold]

I just want to know what this statement means: "all the squares in the multiplicative group $\mathbb{Z}_n^*$."
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0answers
32 views

$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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1answer
28 views

I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial [on hold]

I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial, elementary abelian normal subgroup of $P$ which has a complement $X$ in $P$. If $P = \langle y \rangle X$ for ...
3
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1answer
43 views

Group of order $pqr$ and cyclic subgroup

Let $G$ be group of order $pqr$, when $p,q,r$ are different prime numbers. Does $G$ must have normal cyclic subgroup $H$ such that $G/H$ is cyclic too ? I know that $G$ has normal sylow subgroup of ...
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1answer
23 views

Mordell's theorem-Finitely generated abelian group

In my lecture notes we have the following: Mordell proved the following: Let $C$ be a nonsingular cubic curve with rational coefficients. Then the abelian group of rational points on $C$ is ...
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1answer
43 views

centre of a group presentation

having trouble showing that an element belongs to a centre of a group presentation. Let $G = \langle x,y,z\mid x^2=y^3=z^3=xyz\rangle$ I have to show that $ a = xyz$ belongs to the centre of $G$. I ...
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2answers
50 views
+50

Let $H$ be a subgroup of the group $(R, +)$ such that $H$ $∩$ [-1,1] is a finite set containing a non zero element. Show that $H$ is cyclic.

Observations: Since $H$ is a subgroup of $(R, +)$ so $0 \in H.$ If $1 \in H,$ then all positive integers belong to $H.$ But $H$ is closed wrt addition, so the negative integers must belong to $H$ ...
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1answer
26 views

permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. ...
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3answers
41 views

Building a proper homomorphism between groups.

Suppose I have a cyclic group $G$ of order $6$. I want to show that it is isomorphic to $\Bbb {Z}_6$. So $G=\{e,g^2,g^3,g^4,g^5\}=\langle g\rangle$. Can I build a homomorphism $f:G \to \Bbb{Z}_6$ that ...
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29 views

Order of $\frac{2}{3}$+Z in Q/Z

Let Q/Z be quotient group of addive group of rational numbers. Find order of element $\frac{2}{3}$+Z in Q/Z. I tried it by using facts that any G/H of G has induced operation from G. So I can do ...
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2answers
45 views

If $ H , K$ are subgroups of $G$ and $HK$ is subgroup of $G$ then $|H|$, $|K|$ aren't co-prime?

The statement I'm trying to understand is as written in the title : If $ H , K$ are subgroups of $G$ and $HK$ is subgroup of $G$ then $|H|$, $|K|$ aren't co-prime? I tried to find a counter ...
2
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1answer
25 views

Looking for example of a surjective homomorphism on $(\mathbb R,+)$ which is not an automorphism

Give example of a surjective function $f:\mathbb R \to \mathbb R$ such that $f(x+y)=f(x)+f(y) , \forall x,y \in \mathbb R$ but $f$ is not injective . I think I have to do something with basis of ...
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3answers
30 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
2
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1answer
34 views

Does pigeonhole principle apply for all groups?

I'm reading Rosen's book and it has a proof to show that a finite subgroup (set) is closed under a composition law. It says for some $i$ and $j$, $i < j$, $a^i = a^j$ i.e, $a^i = a^i \circ ...
0
votes
1answer
29 views

class equation of order $10$

Is it a class equation of order $10$ $10=1+1+1+2+5$. As far as I know for being a class equation each member on RHS has to divide $10$ and should have at least one $1$ on RHS, which is ...
2
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0answers
29 views

A question about group actions on a trees [on hold]

why does the following conclusion hold: Let G be a group acting on a tree $\Gamma$, H a subgroup of G with minimal subtree $\Gamma_H$ and $g\in G$ be a hyperbolic element, s.t. $\langle g\rangle\cap ...