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

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3
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prove or disprove isomorphism in group theory

In group theory to prove two groups are isomorphic we have to prove that there exist one-one onto map and also operation is preserved. For example if we have to disprove that U(10) and U(12) are not ...
0
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1answer
27 views

$m'$-group being cyclic?

Given a group $G=\mathbb{Z}_m\rtimes\mathbb{Z}_n$ with $m,n$ coprime. Should every subgroup of $G$ that has order coprime to $m$ be cyclic?
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5answers
155 views

Prove the following about $S_n$ and $A_n$

Prove that $\sigma \tau \sigma^{-1} \tau^{-1} \in A_{n}$ for any $\sigma,\tau \in S_{n}$ So I was thinking that since $A_{n}$ is a subgroup of $S_{n}$ that you can just rearrange $\sigma \tau ...
5
votes
1answer
303 views

Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$

Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
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3answers
228 views

Permutation and group isomorphism question.

$S_3$ is isomorphic to $D_3$ using an equilateral triangle. Similarly, the book uses a square to represent $D_4$. Now, I know the geometries (rotations, reflections) on this square which comes up to ...
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2answers
67 views

Let $\sigma = (15793)(2468) ∈S_9$ Find all $τ ∈ S_9$ such that $τ^3=σ$. And prove there are no more such $τ ∈S_9$.

Let $\sigma = (15793)(2468) ∈S_9$ Find all $τ ∈ S_9$ such that $τ^3=σ$. And prove there are no more such $τ ∈S_9$. I found $τ=(17359)(8642)$. Is this the only solution ? Or are there more solutions ...
2
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1answer
191 views

Cyclic subgroup of dihedral group

Let $D_n$ be the dihedral group of order $2n>4$; so it contains a cyclic subgroup $C$ of order $n$ on which $\sigma \in D_n$ outside of $C$ acts as $\sigma c \sigma^{-1}=c^{-1}$ for all $c \in C$. ...
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3answers
496 views

How to find subgroups of $ \;\;\Bbb Z_2\times \Bbb Z_6$

I am reading a first course in algebra and there is an example saying that "find all the subgroups of $\Bbb{Z}_2\times\Bbb{Z}_6$ and decide which of them are cyclic. I know that ...
5
votes
5answers
177 views

Let $x,y$ in a group G with uneven order. Let $x^2=y^2$. Show that $x=y$.

Let $x,y$ in a group G with uneven order. Let $x^2=y^2$. Show that $x=y$. Okay, how do I need to prove this? If $G$ is unenven order, then there are no elements of order $2$. So $x^2=y^2=e$ only if ...
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2answers
185 views

Prove that k-cube graph is a Cayley graph

Define Cayley graph as following: G is finite group. C $\subseteq$ G such that C does not contain identity element of G and g-1 $\in$ C for all g $\in$ C. Cayley graph X(G,C) is formed with vertices ...
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1answer
96 views

How to form Cayley graph from group

Define a Cayley graph as follows: $G$ is finite group. $C \subseteq G$ such that $C$ does not contain identity element of $G$ and $g^{-1} \in C$ for all $g \in C$. Cayley graph $X(G,C)$ is formed ...
5
votes
1answer
161 views

double coset, stabilizer and orbits

$G$ is a finite group. $H\le G$, $D$ is a set. $x \in D$. $G$ acts transitively on $D$. Show that the mapping $HgG_x \mapsto Hgx$ is a well-defined bijection from the set of double cosets $\{HgG_x: ...
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0answers
99 views

Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
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2answers
56 views

Prove that $H$ is generated by the elements $(1,2)$ and $(0,3)$.

Let $H:= \{(x,y) ∈ ℤ×ℤ : 5x-4y≡0 \bmod{6} \} $. Prove that $H$ is generated by the elements $(2,1 )$ and $(0,3)$.
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1answer
79 views

Is this reasoning correct?

Let G be a finite group of uneven order. Let $x \in G$ be an element with the property that there exist an element $g \in G$ such that $gxg^{-1} = x^{-1}$. Prove $x$ is the idendity of $G$. Here is ...
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1answer
521 views

Relation between semidirect products, extensions, and split extensions.

When I read the textbook about semidirect products and split extensions...I feel like I'm lacking the intution behind them and also the relation between these two. I was wondering if anybody could ...
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0answers
118 views

Show group is isomorphic to finite Heisenberg group

Show that the group $\langle x,y,z$ $|$ $z = xyx^{-1}y^{-1}$, $zx=xz$, $zy = yz$, $x^n = \mathbb{I}, $ $y^n = \mathbb{I}$, $z^n = \mathbb{I} \rangle$, $(n \in \mathbb{Z_{>0}})$ is isomorphic to ...
2
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0answers
65 views

Show that if $1+\frac {1}{2}+\frac{1}{3}+\cdots +\frac {1}{p-1}=\frac {a}{b}$then a is divisible by $p^2$ [duplicate]

Here $p$ is a prime number, grater than 3. I can not understand how do I start.Actually I found this problem from Herstein of page 112.I need some hints plz.. Edit : The original post said ...
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2answers
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Action of $\text{SL}_2\mathbb{C}$ on $\mathbb{C}^3$ induces a 2:1 covering $\text{SL}_2\mathbb{C}\to \text{SO}_3\mathbb{C}$

Exercise 7.17 in Fulton's Representation Theory reads, Identify $\mathbb{C}^3$ with the space of traceless matrices in $M_2\mathbb{C}$ so that $g\in \text{SL}_2\mathbb{C}$ acts by $$A\mapsto ...
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1answer
161 views

why do most finite groups of order 128 resemble (at a distance) the elementary abelian group?

As a result of this previous question, I made the following video: Cayley Tables of All Groups of Order 128, and what is striking is that most of them, if you squint, kind of resemble the elementary ...
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3answers
98 views

Some groups of order $40$

Is there some table on the web giving information about particular small groups, that would go up to order $40$ and that would give enough information so that one could be sure whether groups matching ...
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1answer
61 views

Direct sum of a group and generator

When talking of a group $G$ being the direct sum of a set of subgroups $\{H_i\}$, the following part is a part of definition (from wiki): $G$ is generated by the subgroups $\{H_i\}$ So, does this ...
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1answer
382 views

right group action

wikipedia says 'The difference between left and right actions is in the order in which a product like $gh$ acts on $x$. For a left action $h$ acts first and is followed by $g$, while for a right ...
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votes
3answers
130 views

Order of $D_4$ is $4$ or $8$?

$D_4 = \{R_0,R_{90},R_{180},R_{270},H,V,D,D^{'}\}$ $|R_0|=1 ,\, |R_{90}|=4 ,\, |R_{180}|=2 ,\, |R_{270}|=4$ $|D|=|V|=|D|=|D^{'}|= 2$ So Does that mean order of $D_4$ is $4 = \text{lcm}\, (4,2,1).$ ...
3
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1answer
194 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
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2answers
240 views

Order of permutation

What does order of permutation means??? and how to prove that the order of permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles
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1answer
242 views

Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
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1answer
441 views

Intersection of cosets

I'm not really sure how to go about proving the following. Any help will be appreciated. Question: Let $H$ and $K$ be subgroups of a group $G$. Prove that the intersection $$xH \cap yK$$ of two ...
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2answers
39 views

Extension of a finite group

I am working on extension of finite groups. Let $G$ be a finite group with trivial center such that it has a normal subgroup $N$ with odd order and $G/N=L_{2}(p)$ ($p$ is prime). I am looking for an ...
5
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1answer
222 views

Character theory exercises [closed]

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them. In particular, I would need help with these ones. Thank you very much ...
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2answers
310 views

Let $G$ be a finite group and $P$ be a Sylow $p$-subgroups of $G$. Let $H$ be a subgroup of $G$ such that $N_G(P)\subset H$. Prove that $N_G(H)=H$

1.Let $G$ be a finite group and $P$ be a Sylow $p$-subgroups of $G$. Let $H$ be a subgroup of $G$ such that $N_G(P)\subset H$. Prove that $N_G(H)=H$ 2.Let $G$ be a group of order $143$. Show that ...
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2answers
149 views

a powerset define a group?

Let $P(E)$ denote the power set of a set $E$: the set of subsets of $E$. Does the operation $A\cap B$ define a structure of group? By denition, a group $G$ is a set with an operation $g.h$ (formally, ...
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1answer
174 views

How can I write the product of two transpositions as product of 3-cycles?

Consider this equality in the group $S_n$ : $$(ab)(cd)=(cad)(abc)$$ I can prove it for myself, but I get the impression that I should be able to see this very quickly. How could I get more intuition ...
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1answer
63 views

Example of a finite group

I need an example of a finite group $G$ such that 1) the number of Sylow $p$-subgroup of $G$ is $1$ for all $p\neq 2$ 2) the number of Sylow $2$-subgroup of $G$ is $3$.
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2answers
33 views

Showing that the set of all left multiplications is the centralizer of the set of right multiplications and vice versa

Given a group $G$, let $L$ and $R$ be the subgroups of $S=\operatorname{Sym}(G)$ consisting of all left multiplications by elements of $G$ and all right multiplications by elements of $G$, ...
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1answer
44 views

totally ordered group

Suppose a no trivial totally ordered group .This group has maximum element? A totally ordered group is a totally ordered structure (G,∘,≤) such that (G,∘) is a group.I couldnt find a more exact ...
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3answers
160 views

Order of the quotient group $\;\mathbb R^*/G^*$?

Let $\mathbb R^*$ be the group of all non zero real number under multiplication and $\,G^*$ be the subgroup of $\mathbb R^*$ consisting of all squares of reals. What is the order of the quotient ...
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2answers
331 views

Nonabelian group of order $p^3$ and semidirect products

Let G be a nonabelian group of order $p^3$ where p is an odd prime. Suppose that G contains an element of order $p^2$. Then G is isomorphic to the semidirect product $Z_{p^2} \rtimes_{\alpha} Z_p$, ...
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How do I find $\left|\langle a,b\mid a^2=b^3=e\rangle\right|$?

Suppose $G$ is a group satisfying $G=\langle a,b\mid a^2=b^3=e\rangle$. Find $|G|$.
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Please help me, Group Theory. Prove $b^{33}=e$.

Let $G$ be a group and $a,b \in G$. Prove that if $a^{2}=e$ and $ab^{4}a=b^{7}$, then $b^{33}=e$, where $e$ is the identity of a group $G$.
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1answer
113 views

Quotient group as colimit

I have been wondering for a while about the following question without getting anywhere: Let $G$ be a group, $N$ a normal subgroup. Can the quotient group $G/N$ be seen as the (category theoretical) ...
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3answers
102 views

Subgroups of $\mathbb{Z} \times \mathbb{Z}_2$

Is it true that the only non-trivial (subgroups other than the trivial group and the group itself) subgroups of $\mathbb{Z} \times \mathbb{Z}_2$ are all isomorphic to $\mathbb{Z}$? My intuition tells ...
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4answers
343 views

Is the free group on an empty set defined?

I'm guessing that the free group on an empty set is either the trivial group or isn't defined. Some clarification would be appreciated.
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4answers
691 views

Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic?

Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic under addition? And how could I prove this ? What about $\mathbb{Q}$ and $\mathbb{Q}[i]$ ?
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1answer
114 views

Can we find subgroup of $(\mathbb{R},+)$ with order 2?

We used the following idea: first get a set of Hamel basis for $\mathbb{R}$, secondly, divide it into two parts such that one set of the Hamel basis forms a group, the other one is just the former one ...
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2answers
94 views

Property of abelian group

I need to show the following: If G is abelian group of order n, then $f(x)=x^m$, where (n,m) are co-prime, is an automorphism of G I know it needs Lagrange theorem, but would appreciate some ...
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5answers
725 views

Understanding Lagrange's Theorem (Group Theory)

I am beginning with Abstract Algebra and I'm trying to understand Lagrange's Theorem. The theorem reads For any finite group $G$, the order of every subgroup $H$ of $G$ should divide the order of ...
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2answers
132 views

$G$ a group where $H$ is a subset of $G$ with index $[G:H]<+\infty$

$G$ is a group where $H$ is a subgroup of $G$ with index $[G:H]< +\infty$. Show if for the element g is an element in G we have $gHg^{-1}$ is a subset of H. Then we must have$ gHg^{-1}=H$. (This ...
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5answers
366 views

what $D_8/D_8$ is isomorphic to

What is $D_8/D_8$ isomorphic to? Why $\,1\,?\,$ By definition, I think it should be $D_8$, since $\,D_8/D_8=\{gD_8|g\in D_8\}=D_8$. $(D_8/D_8:\;$ the quotient group of $D_8$ by itself.)
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1answer
58 views

Group representation scalar product

Let $\rho: G \rightarrow GL(V)$ be a finite dimensional complex representation of the group $G$. Show that there is an inner product on $V$ such that $G$ acts by unitary matrices. My approach so far ...