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

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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 ...
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17 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 ...
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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 ...
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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 ...
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27 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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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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26 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 ...
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15 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 ...
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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 ...
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16 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 ...
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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 ...
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17 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 ...
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1answer
14 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 $, ...
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1answer
8 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 ...
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51 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. ...
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49 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 ...
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35 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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15 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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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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40 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.
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1answer
20 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 ...
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56 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) = ...
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22 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 ...
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2answers
94 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 ...
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0answers
38 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 ...
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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 $H$,$K$, ...
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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.
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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 ...
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1answer
39 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
32 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 ...
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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 ...
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Unitary groups deals with matrices how it linked forms ??? [on hold]

But Projective Linear groups and all those things are defined on forms.. Is there any connection between forms and matrices... ??
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1answer
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$ |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 ...
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1answer
50 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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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 ...
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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.${}$
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1answer
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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, ...
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1answer
48 views

Abelian isomorphic group [duplicate]

Prove that if $G$ and $G'$ are isomorphic groups and $G$ is abelian, then $G'$ is abelian, too. Can you please solve this question, I have an exam soon and I have to learn this! I know I asked this ...
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Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.

I've tried proving that $ghg^{-1}\in H$ ($\forall g \in G$), but I don't see how the special property of $H$ guarantees this. Any insight? I've turned away from it to work on other things, and it's ...
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Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
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1answer
67 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
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Normal subgroup, quotient group, isomorphism.

Let $R^{*}$ be the group of nonzero real numbers under multiplication and let $R^{+}$ be the group of positive numbers under multiplication. Prove (a) $\{-1,1\}$ is a normal subgroup of $R^{*}$. (b) ...
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Show that if the prime $p$ divides $|G|$, then $|X|$ is divisible by $p$.

Question : Let $p$ be a prime number that divides the order of the finite group $G$. Let $X$ = $\bigcup_{P \in Syl_p(G)}P$. Show that $|X|$ is divisible by $p$.
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Understanding something about divisible subgroups and generated subgroups

So, that means that $dG$ is the intersection of all divisible subgroups of $G$ (that's the definition I know.) But then they say : Here, I think they're saying that $dG$ is the sum of some ...
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show that there is some element x∈X whose stabilizer Gx is all of G where G is a group of order p^k, where p is prime and k is a positive integer

I'm having trouble with this problem: Suppose that G is a group of order p^k, where p is prime and k is a positive integer.
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41 views

Cylic group of order 2

I need to check $\dfrac{D_{4}}{N}$ is isomorphic to the cylic group of order $2$. However, I just want to check if $\mathbb{Z}_2 $is the cyclic group of order 2 since it is not specified. Thank you.
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1
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28 views

Order of the elements of a group of order $3$

Let $S$={$5$,$1/5$,$1$} be a set then I think $(S, .)$ is a group where identity element is $1$.Here order of the group is $3$.What is the order of the elements i.e $5$ and $1/5$? We know that the ...
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1answer
18 views

Automorphism of a group G

Let $G$ be a group, and let $a$ be a fixed element of $G$. Define the function $\gamma_a(x)=axa^{-1}$ for all $x \in G$. Prove that $\gamma_a$ is an automorphism. For $G=S_3$, compute $\gamma_{(1 \ 2 ...