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

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Prime factors of binomials

Is it true that for each $n\geq 2$ there are two primes $p, q$ such that (at least) one of them divides $\binom{n}{k}$ for each $1\leq k\leq n-1$? Examples: For $n=6: \binom{6}{1}=6; ...
1
vote
1answer
30 views

How to stop a calculation in GAP without closing GAP?

I have the following question concerning GAP: If I want to stop a calculation, how can I do this? I know the command Ctrl+Z, but then GAP is closed. I am using Linux Ubuntu 14. Thanks for the ...
1
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0answers
41 views

If $X$ is an Abelian group, then $\ker_X : \mathrm{Cong}(X) \rightarrow \mathrm{Sub}(X)$ is a bijection. Is there a partial converse?

(All monoids are written additively in this question, even the non-commutative ones.) Given a monoid $X$, write $\mathrm{Sub}(X)$ for the lattice of submonoids of $X$, and write $\mathrm{Cong}(X)$ ...
-1
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2answers
56 views

More questions regarding example I. N. Herstein's *Topics in Algebra

I was reading I. N. Herstein's Topics in Algebra and had confusion in the same example as the following post: Confused by Example in Herstein's "Topics in Algebra" I, however, want to ...
0
votes
2answers
26 views

Finding homomorphisms

Let $G$ be the dihedral group of order 14. In the following, justify your answer. 1. Let $A=C_2$ be a cyclic group of order 2. Find all homomorphisms $G→A$. 2. Let $B=C_7$ be a cyclic group of ...
4
votes
2answers
61 views

permutation group, lie group

Let $S$ be any set, and denote by $G$ the collection of all subsets of $S$. For $A, B \in G$ let be $AB = (A - B) \cup (B - A)$. I know how to show that this set $G$, with this product operation is a ...
6
votes
2answers
243 views

Dense uncountable proper subgroup of $(\mathbb{R},+)$

Probably someone had asked this question on StackExchange, but can one construct a dense uncountable proper subgroup of $(\mathbb{R},+)$?
0
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0answers
99 views

Using the Third Isomorphism Theorem

Here's my question Using the Third Isomorphism Theorem, show that if m, n are positive integers then there is an isomorphism: $\Bbb Z_m \cong \Bbb Z_{mn}/\Bbb Z_{n}$ I began this by ...
1
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1answer
27 views

The influence of the finiteness of a set on the conjugation classes of a group

Let $G$ be a torsion group. Suppose that $G = \langle a,B \rangle$ where $a \in G$ and $B$ is a abelian subgroup of $G$. Denote by $a^B$ the set of the conjugates of $a$ by elements of $B$, i.e., $a^B ...
3
votes
1answer
47 views

Quotient of matrix group fails in GAP

This is a question about quotients of matrix groups in GAP… The matrix group generated by a := [[1,1],[1,-1]]/Sqrt(2); b := [[1,0],[0,E(4)]]; G := Group(a,b); ...
0
votes
2answers
47 views

Is there any non-nilpotent group such that every subgroup is self normalizing?

Is there any non-nilpotent group $G$ such that for any proper nontrivial subgroup $H$, $$N_G(H)=H\ ?$$ Edit: Thanks to "ahulpke" and "Myself", We see that it is not possible for finite groups. Is ...
3
votes
3answers
57 views

A group action proof without group actions?

I am currently teaching an undergraduate abstract algebra course out of Saracino, Abstract Algebra: A First Course. Exercise 13.13 asks the following: Let $K$ be the subgroup $\{ e, (1, 2)(3, 4), ...
4
votes
1answer
147 views

A theorem from the theory of groups

Let $K$ be a (not necessarily normal) subgroup of the group $G$ : $K < G$ A fixed element $g\in G$ can act, from the left, on all elements of $G$, thus generating a bijection of $\,G\,$ onto ...
2
votes
0answers
29 views

Decomposiom of the representation of $SU(N)$

Let $T$ be the "fundamental" representation (I mean the one in which the matrices representing the group elements are simply themselves) of $SU(N)$ group. I have \begin{pmatrix} SU(N-1)& 0\\ ...
0
votes
2answers
49 views

Prove every subgroup S of a finitely generated abelian group G is itself finitely generated.

Call a group G finitely generated if there is a finitely subset X$\subseteq$G with G=$<X>$. Prove that every subgroup S of a finitely generated abelian group G is itself finitely generated. I ...
4
votes
1answer
83 views

Is there a homomorphism from a full product of finite cyclic groups onto $\mathbb Z$?

Trying to answer this question, I encountered the following question, the answer to which should be known but it is hard to Google, so I did not find it. Let $G=\prod_{n\in\mathbb N}\mathbb Z_n$ be ...
2
votes
1answer
37 views

Group Action transitivity on fixed point

Given a group G that acts on X transitively, P a Sylow p-subgroup of G and N the normalizer of P, define Y to be the subset of X whose points are all fixed by elements of P. How can I show that N acts ...
0
votes
0answers
30 views

Sylow Theorems for Symmetric (Permutation) Groups

The General Linear Group $GL(n,\mathbb{F}_p)$ has an interesting property that the proof of Sylow theorem for this group can be given which is based on the natural action of this group on the ...
1
vote
1answer
70 views

Could someone check my work on this exercise

I solved the following exercise, could someone please check my work? Exercise: Let $$ A = \left ( \begin{array}{cccc} 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 ...
4
votes
1answer
92 views

Primitive permutation group with subdegree 4

What are the possible sizes of a point stabilizer in a primitive permutation group, where the point stabilizer has an orbit of size 4? In the tradition of subdegree 3 and subdegree 2, I wonder ...
2
votes
1answer
23 views

Polygons with fixed number of axes of symmetry

Let $k$ be a positive integer. Suppose a polygon has exactly $k$ axes of symmetry. How many sides may the polygon have? A regular $n$-gon has $n$ axes of symmetry, so one answer is $k$. What are ...
1
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0answers
29 views

Group presentation and Smith normal form

Problem Let $A=\langle a,b: a-3b=0,3a=3b \rangle$ and $B=\mathbb Z^3/S$ with $S=\{m \in \mathbb Z^3: m_1+2m_2+m_3=0, 5|m_3 \}$. Calculate $\operatorname{Hom}_{\mathbb Z}(A,B)$. My attempt at a ...
0
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1answer
32 views

Conjugacy classes of $S_n$ under the action of $S_{n-1}$

I try to get explicitly сonjugacy classes of $S_n$ under the action of $S_{n-1}$. I believe that in the description of the classes present cycle type of a permutation and yet another parameter. But I ...
4
votes
1answer
689 views

How do I show that every group of order 90 is not simple?

Can comeone help me? How do I show that every group of order 90 is not simple?
1
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1answer
33 views

Non-existence of non-abelian simple groups of order $90$ by counting elements [duplicate]

I want to show that no group of order $90=2 \cdot3^2 \cdot5 $ is simple. Part (iii) of Sylow's Theorem gives $$n_2 \in\{1,3,5,9,15,45\} $$ $$n_3 \in \{1,10\} $$ $$n_5 \in \{1,6\} $$ I am aware of ...
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2answers
47 views

Trying to understand group presentations using the example of the Dihedral group

According to Wikipedia the Dihedral group $D_n \cong \; \langle r,s \mid r^n = 1, s^2 = 1, s^{-1}rs = r^{-1}\rangle$. But why does this apply? As far as I understand the group presentation means that ...
1
vote
2answers
65 views

An example of free group

Let $\alpha : \mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $\alpha(x)=x+2$ and $\beta:\mathbb{C}\cup\{\infty\} \to \mathbb{C}\cup\{\infty\}$ with $ \beta(x)=x/(2x+1)$. Show that the ...
1
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2answers
33 views

Find an element of order $45$ in the group $\mathbb{Z}_{30}\oplus\mathbb{Z}_{12}$, or explain why it is impossible

I'm asked to find the object asked for, or explain why it is impossible. Any help would be appreciated. Thanks for your time.
7
votes
3answers
422 views

Geometrical meaning of automorphisms of cyclic groups

I'm looking for a geometrical interpretation of the action of automorphisms of cyclic groups. I'll take one particular example to make it clear : I'm taking the cyclic group $\mathbb{Z}_{12}$, which ...
1
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0answers
25 views

If $\sigma$ is a cycle of length $r$, then it has order $r$?

I should specify that $\sigma \in S_n$, the symmetric group. I've written down some permutations and it seems like their order correspond to their lengths. How can I prove this? I was thinking of ...
7
votes
1answer
81 views

$GL_n(\mathbb{F})$ contains a copy of $\mathbb{F}^{n-1}$

It is a fact of matrix multiplication that $$\left( \begin{matrix} 1 & a & b \\&1&\\&&1 \end{matrix} \right) \left( \begin{matrix} 1 & a' & b'\\&1&\\&&1 ...
1
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1answer
34 views

Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
0
votes
1answer
22 views

Is the restriction of the regular representation of a finite group always a multiple of the subgroup?

For an inclusion of groups $H \hookrightarrow G$, define the restriction $\operatorname{Res}^G_H$ of representations as precomposition with the inclusion map. Also, define the complex regular ...
1
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0answers
25 views

Let $p,q$ and $r$ be positive prime numbers. Determine the number of abelian groups of order $p^6q^3r$

Let $p,q$ and $r$ be positive prime numbers. Calculate the number of non isomorphic abelian groups of order $p^6q^3r$. I've tried to use the structure theorem. So we have $$G \cong \mathbb Z/\langle ...
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0answers
18 views

Similar transformation matrix restricting determinant to be 1.

How do you prove that if restricting the determinant of a similar transformation matrix between two equivalent irreducible unitary representation of a finite group to be 1, then this transformation ...
0
votes
3answers
70 views

$G/Z(G)$ is cyclic useful for proving groups abelian?

It's a common exercise to prove in an abstract algebra book that if $G/Z(G)$ is cyclic then $G$ must be abelian. But I've always found the exercise strange because if $G$ is abelian then $Z(G)=G$ and ...
2
votes
0answers
23 views

About the Radicable Part of Subgroups of Chernikov Groups

Let $G$ be a (infinite) Chernikov group. Supose that $H$ is a (infinite) subgroup of $G$. Denote by $G^0$ the radicable part of $G$, ie, how $G$ is a Chernikov group then $G$ has a subgroup of index ...
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2answers
37 views

Composition series of nilpotent group

I found this problem in The Theory of Groups by Marshall Hall. Let the group $G$ be of order $p^rq^s$. If $G$ has two composition series $1 \unlhd A_1 \unlhd A_2 \unlhd \cdots \unlhd A_r \unlhd ...
0
votes
1answer
26 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
0
votes
1answer
37 views

Algebra - Group Theory help… Intersection notation

How do you write $\{g ∈ G : \mu (Hx,g) = Hx,∀x∈G \}$ in intersection form? where $\mu (Hx,g) = Hgx$
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votes
2answers
46 views

identify quotient group

Identify quotient group $\mathbb R^*/\mathbb R^+$ where $\mathbb R^*$ is multiplicative group of non zero reals and $\mathbb R^+$ denote subgroup of positive real numbers.I m using first isomorphism ...
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1answer
57 views

Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
1
vote
1answer
30 views

Let $\varphi:G\rightarrow H$ be a homomorphism, show $\varphi':G\rightarrow\text{im}(\varphi)$ is an epimorphism

Let $\varphi:G\rightarrow H$ be a homomorphism, show $\varphi':G\rightarrow\text{im}(\varphi)$ is an epimorphism Epimorphism is a surjective homorphism. We know that $\text{im}(\varphi)\subseteq H$ ...
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votes
2answers
39 views

Group theoretical proof of $\varphi(rs)=\varphi(r)\varphi(s)$ through generators of the group.

Given a group $G=\langle a\rangle$ of order $rs$, with $(r,s)=1$, I showed there exist unique $b,c\in G$ such that $a=bc$ with $b$ of order $r$ and $c$ of order $s$. The latter is a direct consecuense ...
11
votes
3answers
145 views

number of distinct ways of writing each element of the set $HK$ in the form $hk$

Let $H$ and $K$ be subgroups of the group $G$. The number of distinct ways of writing each element of the set $HK$ in the form $hk$, for some $h \in H$ and $k \in K$ is $|H \cap K|$. My thoughts:- ...
1
vote
1answer
25 views

Existence of a generator over multiplication for integers modulo p

If we consider the integers modulo a prime $p$, then for every $x \not \equiv 0$ (mod $p$), we can get any $b \not \equiv 0$ by adding $x$ a number of times to itself. Is the same true for ...
3
votes
1answer
44 views

No character theory, any representation ${\bf{GL}}_N(\mathbb{C})$ is reducible, upper bound

Let $G$ be any finite group. How do I show without character theory that there is a number $N = N(G)$ so that any representation $\rho: G \to {\bf{GL}}_N(\mathbb{C})$ is reducible (and finding an ...
15
votes
1answer
244 views

endomorphism of finite groups

Have $\mathcal{G}$ denote the set of finite groups with at least $2$ elements. How would I go about showing that if $G \in \mathcal{G}$, then $\left|\text{End}(G)\right| \le ...
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0answers
16 views

Invariant factors of finite abelian group

Calculate the invariant factors of the group $G=\mathbb Z_{12} \oplus \mathbb Z_{21} \oplus \mathbb Z \oplus \mathbb Z \oplus \mathbb Z_{20} \oplus \mathbb Z_{9} \oplus \mathbb Z_7$. Applying the ...
0
votes
0answers
39 views

$G$ finite abelian group, $p$ prime that divides order of $G$

Problem Let $G$ be a finite abelian group and $p$ a positive prime that divides $|G|$. Show that the number of elements of order $p$ in $G$ is coprime with $p$. Let $|G|=p^nm$ with $n \geq 1$ and ...