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

learn more… | top users | synonyms (2)

5
votes
1answer
27 views

group iff $G$ is a set, $*$ associative binary operation satisfying left identity, left inverse

How would I go about showing that the pair $(G, *)$ is a group if and only if $G$ is a set and $*$ is an associative binary operation on $G$ such that: (Left Identity) There exists an element $e \in ...
1
vote
0answers
39 views

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ . What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ...
2
votes
1answer
28 views

What are all possible actions by automorphism of $H = \Bbb Z/3\Bbb Z$ on $N = \Bbb Z/6\Bbb Z$?

So the question is "What are all possible actions by automorphism of H on N?" with H = Z/3Z and N = Z/6Z. I completely guessed my way through how to go about solving this, but I started with finding ...
2
votes
5answers
97 views

Why $\mathbb{Z}/n\mathbb{Z}$ isn't a subgroup of $\mathbb{Z}$

Could anyone explain to me why this isn't true? It's listed as an example in our textbook but no reason is given. I've checked the properties of a subgroup, and it seems to follow them. What am I ...
1
vote
1answer
28 views

Evenness and oddness of group code weights

I'm doing exercises in Charles C. Pinter's book A Book of Abstract Algebra and I'm unable to solve problem 7 in section H of chapter 5 (subgroups). I think that there is a solution on this site but ...
2
votes
1answer
98 views

Select k no.s from 1 to N with replacement to have a set with at least one co-prime pair

Given $1$ to $N$ numbers. You have to make array of $k$ no.s using those no.s, where repetition of same no. is also allowed, such that at least one pair in that chosen array is co-prime. Find no. of ...
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 ...
3
votes
0answers
75 views

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
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 $\mathrm{Cong}(X)$ ...
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 ...
0
votes
1answer
56 views

Proper action and compactness

Proposition: Suppose a group $\Gamma$ acts properly by isometries on a metric space $X$. If the action is cocompact then every element of $\Gamma$ is a semisimple isometry of $X$. (Please refer ...
5
votes
0answers
175 views
+100

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
2
votes
2answers
46 views

A group in between the commutator subgroup and the original group must be normal

Let $C$ be the commutator subgroup of a group $G$. then by some easy arguments, we know that $1$. $C$ is normal in $G$ $2$. $G/C$ is abelian $3$. If $N$ is normal in $G$ and $G/N$ is abelian, then ...
0
votes
2answers
54 views

Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive?

Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
5
votes
1answer
47 views

Group of order $p^{n}$ has normal subgroups of order $p^{k}$

Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$. Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$. Suppose it ...
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 ...
-1
votes
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 ...
1
vote
2answers
47 views

What are the normal subgroups of $S_4\times S_3$

What are the normal subgroups of $G=S_4\times S_3$. I think $G$ has a normal subgroup of order $72$ is there any other nontrivial normal subgroups of $G$.
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 ...
-1
votes
1answer
74 views

How to find out whether a group is Abelian [closed]

Let $G$ be the set $\mathbb R\setminus \{0\}$ and let $*$ denote a binary operation on the set, defined by:$$\forall a, b \in G,\;\;\;a*b= \dfrac{a\cdot b}2.$$ I need to show that $[G,*]$ is an ...
1
vote
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 ...
2
votes
0answers
30 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\\ ...
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), ...
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
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 ...
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 ...
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 ...
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 ...
1
vote
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
votes
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 ...
1
vote
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 ...
1
vote
2answers
34 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.
1
vote
2answers
66 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
vote
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 ...
0
votes
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 ...
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 ...
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 ...
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 $\mu (Hx,g) = Hgx$
-4
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 ...
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 $\text{im}(\varphi)\subseteq H$ ...
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 ...
1
vote
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 ...
0
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 ...
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 ...
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 ...
1
vote
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 ...
1
vote
4answers
122 views

How many ways can the vertices of an equilateral triangle be colored using three different colors?

Its not $3^3$ because some of the colorings are equivalent. How would I apply Burnside's theorem to this?
1
vote
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
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 ...
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 ...