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

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If G is a finite abelian group and $a_1,…,a_n$ are all its elements, show that $x=a_1a_2a_3…a_n$must satisfy $x^2=e$.

I have already tried with $S_3$, and indeed, the product is $(13)$, and $(13)^2=e$ But what about this: I define + in this way:$45=2$,$26=3$, $1$ is the identity. therefore, $123456=2326=233=3$, ...
1
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0answers
13 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
1
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2answers
15 views

Difference between F-Automorphism and Identity morphism

In reading Shiffrin Abstract Algebra, in the section on Galois Theory, it gives the following definition: For K a field extension of F, a ring isomorphism $\phi:K\to K$ is an F-Automorphism if ...
3
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2answers
50 views

Number of homomorphisms $\mathbb Z_3 \times \mathbb Z_3\to\mathbb Z_9$

I had this wonderful idea: $f$ is homomorphism: $G\to H$, $|G| = |\ker f| \cdot |\operatorname{im} f|$, $\ker f$ - subgroup of $G$, and $\operatorname{im} f$ - subgroup of $H$, so their orders must be ...
2
votes
1answer
60 views

Abstract Algebra Matrix Group Theory

The matrix group G = SL(n, $\mathbb{R}) = \{A \in M(n, \mathbb{R})\} \text{ acts on } X= R^n$ by left matrix multiplication: $\tau _A(x) = A\cdot x (\text{matrix product }(n \times n ) \cdot (n \times ...
2
votes
0answers
25 views

Proof verification that group elements follow law of exponents

I have a proof for the following proposition: Suppose $G$ is a group, $g\in G$ and $m,n \in \mathbb{Z}$. Then $(g^m)^n=g^{mn}$ Proof If $m$ and $n$ are positive then it is clear that ...
1
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0answers
18 views

Discrete $G$-modules and questions on some basic properties (generators, union of its finitely generated submodules)

I am currently stuck with the following proposition: Let $G$ be a profinite group and let $M$ be a $G$-module. (a) If $M$ is a discrete $G$-module, then it is finitely generated as a $G$-module if ...
1
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1answer
42 views

Intertwining map in Schur's Lemma

I am learning Schur's Lemma from page 4 here. It says Schur's Lemma 1. If $(\rho_1, V_1)$ and $(\rho_2, V_2)$ are irreducible representations of a group $G$, then any nonzero homomorphism $\phi : ...
2
votes
1answer
40 views

If every left coset of $H$ is a right coset the show that $H=aHa^{-1}$ for all a in G

$H$ is a subgroup of G. My attempt: $ha=ah' $ for every $h\in H$, where $h'\in H$ doesn't necessarily equal to $h$. So for each $h\in H$, $h=ah'a^{-1}\in aHa^{-1}$, so $H\subseteq aHa^{-1}$. Then how ...
0
votes
2answers
59 views

solutions of $x^2\equiv 1 \pmod p $ [duplicate]

If p is a prime, show that the only solutions of $x^2\equiv 1 \pmod p $ are $x=1$ and $x\equiv -1 \pmod p$. (from herstein's abstract algebra chapter2 section4 lagrange's theorem problem 15, this ...
0
votes
0answers
24 views

$\operatorname{Aut}(H_0)$ isomorphic to $\operatorname{GL}(3,\mathbb R)$?

Is $\operatorname{Aut}(H_0)$ (space of invertible endomorphisms over the space $H_0$) isomorphic to $\operatorname{GL}(3,\mathbb R)$ ? $H_0$ is the space of pure imaginary quaternions (which is ...
0
votes
0answers
23 views

Prove that $U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$

If $m = n_1 n_2 \cdots n_k $ where $\gcd(n_i~,n_j)=1 ~~ \forall i \neq j$, then prove that: $$U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$$ where $\times$ refers to the ...
-1
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0answers
23 views

How to show that there are as many left cosets as there are right cosets? [duplicate]

G is a finite group and H is a subgroup, How to show that there are as many distinct left cosets of H as there are right cosets? (If this is a duplicate, why not show me where is the original one, ...
1
vote
1answer
48 views

How to show that these two groups are isomorphic

I'm having trouble proving that two groups are isomorphic. I am having trouble with both the homomorphisms and the bijections. How would I go about solving this 2 part question: Prove that the ...
10
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2answers
70 views

Embedding as a subgroup

Suppose I am given two finite groups $G$ and $H$ (not too large: let's say their orders are around $10000$ and $100$ respectively, and the order of $H$ divides the order of $G$). These may be ...
1
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2answers
41 views

triangle groups

I am having a hard time finding references(apart from wikipedia) for the geometric interpretation of triangle groups $$T_{a,b,c} =\langle x,y: \, |x|=a,|y|=b,|xy|=c \rangle.$$ How can these groups be ...
6
votes
7answers
125 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
0
votes
1answer
29 views

Let $p$ be a prime number and $G$ a group of order $p^2$. Show that $G$ has at most $p +1$ subgroups of order $p$.

Let $p$ be a prime number and $G$ a group of order $p^2$. Show that $G$ has at most $p +1$ subgroups of order $p$. To be honest, sometimes I tried to read the content, and could not get out of ...
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0answers
36 views

Abstract Algebra: Prove that every field has only trivial ideals [on hold]

Prove that every field has only trivial ideals (that is, {0} and the field itself)
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0answers
42 views

Let G be a finite abelian group. Let p be a prime number.

i) Let $G$ be a finite abelian group. Prove that the product of all elements in $G$ has order $2$. ii) Let $p$ be a prime number. Use (i) to prove: $((Z/pZ)\setminus\{0\}, \cdot )$ only has one ...
0
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1answer
17 views

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders.

Look for the group members $\left ( (\mathbb Z/24\mathbb Z)^*,\underset{24}{\odot} \right )$ and calculate their orders. The elements are $\overline1,\overline2,\ldots,\overline{23}$. Have the ...
2
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1answer
36 views

GR8767 Question 54

Let G be a group, and fix a an element $G$. The function $f$ from $G$ to $G$ defined by $f(x) = axa^2$ is a group homomorphism if and only if A. $G$ is abelian; B. $G = {e_G}$; C. $a = e_G$; D. $a^2 ...
0
votes
1answer
18 views

Evidence about the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$

Be $p$ an odd prime number. Show that the group $\left ( (\mathbb{Z}/p\mathbb{Z})^*,\underset{p}{ \odot} \right )$ has a unique element of order $2$, namely $\overline{p-1}$, and show that ...
1
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1answer
18 views

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$, $|G|=n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$.

Be $G=\{ e,g_1,g_2,\ldots, g_n \}$ an abelian group of order $n+1$. Suppose $G$ has a unique element of order $2$, say $g_1$. Show that $eg_1g_2\ldots g_n=g_1$. I have serious difficulties with ...
1
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0answers
25 views

primitivity of permutation group

If we are to check that extended triangle group $(p,q,r)=<x,y,t: x^p=y^q =t^2=(xy)^r=(xt)^2=(yt)^2=1>$ is primitive, how we can check it in GAP if we have permutation representation of x,y and t ...
0
votes
1answer
17 views

Decompose complex vector by SU(4)

This question is about to decompose (or reduce dimension) complex vector by SU(4). Given any $4\times1$ complex vector $B$. We can build independent matrix $A_i\in SU\left( 4 \right) ,i=1\ldots n $, ...
0
votes
1answer
15 views

The restricted direct product is a normal subgroup of the direct product

Let $G =\prod _{i \in I} G_i $ be the product of $G_i (i \in I).$ Let $ H= \{f \in \prod G_i: f(i) = e_i \text{ for all finite many element of I} \rbrace$. Show that $H\lhd G $ I am genuinely don't ...
4
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0answers
55 views

Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
3
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2answers
56 views

Frattini subgroup of a finite group

I have been looking for information about Frattini subgroup of a finite group. Almost all the books dealing with this topic discuss this subgroup for p-groups. I am actually willing to discuss the ...
0
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0answers
37 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...
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0answers
21 views

Quotient braid group as a representation of SU(n)

I am working with the quotient braid group $B_3 (3) = B_3 / \langle\sigma_1 ^3\rangle$, where I construct a vector space $V$ so that every element $a \in B_3 (3)$ has a corresponding basis vector ...
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1answer
35 views

Classifying $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$

I wish to classify $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$ according to the fundamental theorem of finitely generated abelian groups. We have that it is of ...
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0answers
50 views

Give an example of a group of size four, acting on a set of size four. [on hold]

please help me with examples for i am confused Describe the orbits and stabilizers for the example you gave.
1
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1answer
21 views

prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$.

I want to prove that the maximal torus of $SO(3)$ is the maximal torus of $GL_3(\mathbb{R})$. I want to use the theorem that every maximal torus of G equals $gTg^{-1}$ for some $g \in G$. But I am not ...
1
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0answers
58 views

Show that (Z4;+4) and (Z5*,.5) are Isomorphic groups [on hold]

Given groups (Z4;+4) and (Z5*,.5). Show that these groups are isomorphic by exhibiting a one-to-one correspondence alpha between their elements such that a+b = c (mod 4) iff alpha(a).alpha(b) = ...
0
votes
0answers
23 views

prove splits compatible if and only if edge-split

"Prove that if $e_A$ and $e_B$ are distinct edges of a binary $X$-tree $T$ and $C=A\Delta B$(symmetric difference), then the splits $\sigma(A), \sigma(B)$ and $\sigma(C)$ are compatible if and only if ...
5
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2answers
97 views

Equivalence relation to make a group commutative

A while ago I was wondering if there is a "natural" way to make a commutative group out of an arbitrary one. I played with the idea a bit and here is what I came up with. Define a binary relation ...
2
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0answers
40 views

Suppose that $|G| = p^aq$, where p and q are primes and a > 0, Then $G$ is not simple?

Proof : We can assume that p$ \neq $ q and $n_p >1 $, so $n_p$ = q. Now choose distinct Sylow p- subgroups $S$ and $T$ of $G$ such that $|S\cap T|$ is as larger as possibe and write $D = S \cap ...
0
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1answer
25 views

Ruling out orders when applying Sylow's theorems

Going through examples of applications of the Sylow theorems in Fraleigh's book, when proving that no group of order 36 is simple, after concluding that $| H \cap K | = 3$ for two $3$-Sylows $H$,$K$, ...
0
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1answer
36 views

modular group isomorphic to free product of groups

I am trying to prove that the modular group PSL(2,Z) is isomorphic to the free product of groups Z_2*Z_3. Any ideas on how to get around this? Any hints much appreciated.
1
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1answer
34 views

Representing Groups as matrices

How to represent any group as group of matrices ? Like how to represent dihedral (4) group (order $8$) as group of $2$ by $2$ matrices ? How to represent direct product of $Z_2$ and $Z_2$ as a group ...
2
votes
1answer
40 views

Is my application of Burnside's Lemma correct in this combinatorial problem?

For a course in Combinatorics (I know very little group theory unfortunately), we've been tasked to use Burnside's Lemma on the following problem: Suppose you write a 5-digit number on a piece of ...
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2answers
36 views

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$.

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$. Also another question, do p and q have to be distinct for this to hold? On top of ...
0
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2answers
36 views

Dihedralize Twice - dihedralize a dihedral group $D_n$

A simplest dihedral group $$D_4=C_2 \ltimes C_4$$ can be regarded as dihedralizing a $C_4$ by a semi-direct product. Q: Can one dihedralize the group $D_4$ a second time by defining $$C_2 \ltimes ...
0
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0answers
26 views

Unitary groups deals with matrices how it linked forms ??? [on hold]

Unitary groups are collection of matrices that has certain properties but in some texts projective unitary groups are defined as collection of sesquilinear forms with some conditions,,, how to relate ...
1
vote
1answer
26 views

$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$

I have done part (i) I Was doing part (ii) and got stuck: Since from above I showed that $H_1 \times H_2 \subseteq K$ now i only need to show that $H_1 \times H_2 \supseteq K$. Let $(g_1,g_2) \in ...
1
vote
1answer
59 views

Is there a cayley graph for the Klein bottle?

When studying algebraic topology we learned about the fundamental group of the $2$-torus $T^2$ which is isomorphic to $$\langle a, b \mid aba^{-1}b^{-1} \rangle$$ (the free abelian group on two ...
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votes
0answers
24 views

Prove that G is a group acting on a set X. Where G= {(1),(123),(132),(45),(123)(45),(132)(45)} and X= {1,2,3,4,5}

I understand that the axioms that must be satisfied to prove that this is an "action" is: ex = x for all x an element of X (compatibility with identity). g_1(g_2*x) = (g_1g_2)*x (compatibility with ...
1
vote
0answers
34 views

How to determine the Galois group of a general polynomial over rational number field?

How to determine the Galois group of a general polynomial over rational number field? For example $f(X)=X^n-X-3$, where $n$ is an positive number.${}$
1
vote
1answer
35 views

Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...