A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Can anyone explain these conclusions? Permutations, Symetric group…

The conclusions start off like this:I will highlight what is unclear in yellow. $sgn G$-sign of G permutation, $Ker$-kernel of a function Lets define the function: $\ \Phi$ like: $(\forall G \in ...
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2answers
50 views

Exercise about finding group isomorphisms

So, i've just learned about groups homomorphisms and isomorphisms. I know that an homormorphism betweet two groups is a function such that $$\phi(a\square b) = \phi(a)\star \phi(b)$$ And when the ...
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2answers
52 views

Subgroup of a group with 24 elements.

Suppose that $G$ is a finite group of order $24$, which has four $3$-sylow subgroups. We know that may contain $1$ or $3$ 2-sylow subgroup. How can I prove that there only exists one $2$-sylow ...
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2answers
124 views

Consequence of First Homomorphism Theorem?

Let $\phi:G\to\bar G$ be a surjective homomorphism with kernel $N$. Then the first homomorphism theorem tells us that $G/N\cong\bar G$. My question is this: Lagrange's theorem also tells us that ...
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1answer
106 views

A question on subgroups of a finite group

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$. Prove that $K$ is a subgroup of $H$. So far we found that $o(K)$ divides ...
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38 views

Dihedral subgroups of $S_4$

Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help ...
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1answer
20 views

Counting symmetries using elementary method

I am studying group theory using Armstrong's Groups and Symmetry, one of the biggest problem is that there is no solution manual available. Thus I will rely on you guys! Find all the rotational ...
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4answers
61 views

$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$. [duplicate]

Isn't it obvious for $|a|=|a^{-1}|$, since $\langle a\rangle = \langle a^{-1} \rangle$? For $|ab|=|ba|$, I think we should go like this: $e=(ab)^n\Rightarrow e=(ab)(ab)^{n-1}\Rightarrow ...
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1answer
35 views

Let G be a group, $N<K<G$ and $N\trianglelefteq G$. Prove that $K/N \trianglelefteq G/N$

Let G be a group, $N<K<G$ and $N\trianglelefteq G$. Prove that $K/N \trianglelefteq G/N$ What I have tried is: Note that $1\in G$. So $1\in K$ and $N1=N\in K/N$ which shows that $K/N$ is ...
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98 views

Finite Abelian Groups question [closed]

Let A be a finite abelian group. Let m be the smallest natural number such that ma=0 for every a in A. Prove that there is an element in A of the order m
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33 views

Show that left cosets partition the group

I know how to prove that it happens, by proving that the left coset definition actually is an equivalence relation. Then, it's proved that it partitions the set, since equivalence relations do it. ...
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20 views

A graph and metric on the class of finitely presented groups

Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X \subseteq Y$, $G_1 \cong F\{X\} / H$ and $G_2 \cong F\{X\} / ...
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82 views

Prove that $H$ is a normal subgroup of $G$

This is a problem from the book "Berkeley Problems in Mathematics": Let $G$ be a group of order $120$, let $H$ be a subgroup of order $24$, and assume that there is at least one coset of $H$ ...
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1answer
33 views

The non-functional number

Lets say $f(1)=1, f(2)=2, f(3)=3,$ and $f(4)=n$. Some rules to follow are that: $1)$ $f(x)$ could literally be any function and it depends on what $n$ is. 2) $n$ is an integer. 3) If there is an ...
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1answer
48 views

If $G$ is a group and $a,b \in G $, then: $ |\langle a,b \rangle|=|\langle a\rangle|\cdot |\langle b\rangle| \ \Longleftrightarrow a=b$?

Such that $\langle a\rangle , \langle a,b\rangle $ is a subgroups of $G$ generated by $\lbrace a\rbrace, \lbrace a,b\rbrace $ respectively and $| \ . |$ is the member of $\langle a\rangle $ ...
4
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1answer
76 views

A group satisfying $G=[G,G]$?

What kind of non-identity group $G$ satisfies $G=G'=[G,G]$? How is it related to a solvable or nilpotent group? Thanks in advance.
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28 views

If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots \unlhd G_{s} = G $ is a chief series of $ G $.

The $ p $-Fitting Subgroup of $ G $ is the maximal normal $ p $-nilpotent subgroup Of $ G $ and write it $ F_{p}(G) $. If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots ...
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29 views

Which abelian groups have only a single composition series?

Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$. But cyclic ...
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2answers
34 views

A question about Coxeter groups.

Let $G$ be a group, and $x,y,z \in G - \{1\}$, where $1$ is the identity element of $G$. Assume that $x^2=y^2=z^2=1$, $xy =yz$ and $xz$ is of infinite order. Can $G$ be a Coxeter group? Can $G$ be ...
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1answer
18 views

How to generate the icosahedral groups $I$ and $I_h$?

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their ...
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1answer
19 views

An inductive limit of amenable groups is amenable

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one. Thanks in advance. Edit: I wanted it for ...
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9 views

Weights system corresponding to reflected Dynkin diagram?

Given a set of weights corresponding to the $SO(10) Dynkin diagram How can I transform these weights into weights that correspond to the Dynkin diagram ?
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26 views

Injective homomorphisms and subgroups

I've just started self-learning group theory, and I came accros Cayley's theorem. Which made me wondering about the following question: Given two groups $G$ and $H$, and there exist an injective ...
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3answers
75 views

What does the quotient group $(A+B)/B$ actually mean?

I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements ...
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1answer
55 views

Proof that $\Bbb R/\Bbb Z$ is isomorphic to $S^1$

I just learned the first isomorphism theorem for groups and now I need to prove that: $$\Bbb R/\Bbb Z \cong S^1$$ In other words, the quotient group of $\Bbb Z$ in $\Bbb R$ is isomorphic to the ...
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1answer
53 views

Let $G$ be a group of finite order, $H$ and $K$ subgroups so that $H \unlhd G$; $K \unlhd HK \unlhd G$ and $(|H|,|K|) = 1$. Show that $K \unlhd G$.

I've been trying to solve this for a little while. I know that $H\cap K = \{1\}$ because of their orders and from the isomorphism theorems I know that $ HK /K \simeq H$. I've been trying to see if ...
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1answer
48 views

Suppose $ G = G_{1}G_{2} $. For any prime $ p $, there exist $ P , P_{1} , P_{2} $ such that $ P = P_{1}P_{2} $

Suppose $ G = G_{1}G_{2} $. For any prime $ p $, there exist $ P , P_{1} , P_{2} $ such that $ P = P_{1}P_{2} $, where $ P \in Syl_{p}(G) $ and $ P_{i} \in Syl_{p}(G_{i}) $ , $ i = 1,2 $. The proof ...
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1answer
70 views

How can I prove that this Group is Abelian? [duplicate]

$(G,\cdot)$ a group. If $\exists n\in \mathbb{Z} $ such that $(a\cdot b)^{n+i}=a^{n+i}\cdot b^{n+i}$ for $i=0,1,2.$ $\forall a,b \in G$. Prove that $(G,\cdot)$ is Abelian. I'm not sure how to prove ...
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22 views

A measure of a groups commutativity

Let $G$ be a finitely presented group on and let $ma(G) = n$ be the minimum of $\left| K \right|$ ($K \subseteq G$) over all group presentations $G \cong F\{A\} / H$ such that $G / {H \vee [K, K]}$ is ...
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4answers
50 views

Let $H, K$ be two subgroups of $G$. If $|H| = 12$ and $|K|=17$ then $H \cap K = \{e\}$

My reasoning: Since $|K| = 17$ and $17$ is prime, then any subgroup of $K$ is cyclic. Also, the order of any subgroup must divide the order of the group. But since the subgroups of $K$ must have an ...
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3answers
84 views

Proving that $gHg^{-1}$ is a subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$ with $\operatorname{ord}(H)=k$ I need to prove that $gHg^{-1}$ is a subgroup of $G$ $\color{grey}{(gHg^{-1}=\{ghg^{-1}\mid g\in G, h\in H\})}$ My ...
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152 views

$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p

Let $G$ be a non abelian group of order $p^3$, with $p$ prime. I'm proving that $Z(G)$ (its center) is of order $p$. I already know how to do it by saying that its order can't be $p^3$, nor 1, and if ...
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25 views

If you check Groupisomorphism - does the operation, under which the Groups are closed, matter?

For example: $G_1=(R,+) \quad G_2=(R_+,x)$ So the first group are the Real Number closed under addition and the second group are the positive Real Numbers closed under multiplication. My Lector said ...
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1answer
28 views

Finding subgroups of $\mathbb{Z}_{20}$

I need to find all the subgroups of $\mathbb{Z}_{20}$ My attempt: $\mathbb{Z}_{20}$ is cyclic $\Longrightarrow$ all the subgroups will be also cyclic, according to Lagrangh the order of the ...
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40 views

If $ Q = \langle y \rangle X $ for some element $ y $, then $ \vert N \vert = p $ if $ p $ is odd and $ \vert N \vert \leq 4 $ if $ p = 2 $.

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $ which has a complement $ X $ in $ Q $. If $ Q = \langle y \rangle X $ for some element $ y $, then ...
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1answer
56 views

Universal property of quotient group

Let $H\le G$ be a normal subgroup of $G$. Then we can think a natural projection map $\pi:G \to G/H$. Then this map has the following universal property: Let $\phi:G \to G'$ be a homomorphism. If $H ...
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New matrix operation. Does it have a name? Is it useful?

Matrix multiplication can be defined as: $$({\bf A}{\bf B})_{kl} = \sum_{i=1}^{n} {\bf A}_{ki}{\bf B}_{il}$$ Lately, I have been thinking of another operation: $$({\bf A} \vec{\circ} {\bf B})_{kl} ...
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46 views

Calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$

I'm trying to calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$. I'm really bad with groups so I'd appreciate a verification of my conclusion: If $m$ is even then $\forall x\in \mathbb Z_m$ we get ...
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1answer
57 views

Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$

I need to prove or disprove: Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$ My attempt: No, there isn't isomorphism, because if it did then $S_4$ would have an element of order $8$, ...
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1answer
58 views

There is no injective morphism from $\mathbb S_7$ to $\mathbb A_8$

I am trying to show that it doesn't exist an injective group morphism $f:\mathbb S_7 \to \mathbb A_8$. If there is an injective morphism from $\mathbb S_7$to $\mathbb A_8$ then $\mathbb S_7$ is ...
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1answer
46 views

Replacing $\mathbb{Z}$ with $\mathbb{R}$ in group presentation constructions

Consider a finitely generated group $G$. Assume for a moment it is genearated by $n = 2$ elements. Then the group $G$ has a presentation as a some quotient $\mathbb{Z} \ast \mathbb{Z} / H$. Now ...
3
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1answer
27 views

A free group construction with real (not integral) exponents?

The construction of free groups, on considers "reduced" formal products of letters/symbols with integer exponents. One example of an element in the free group over the letters $\{a, b, c\}$ might be ...
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25 views

Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...
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1answer
58 views

Number of elements of order $11$ in group of order $1331$

Let $G$ be a group of order $1331$. Prove that $G$ has at least $11$ elements of order $11$. $|G|=1331=11^3$ So by First Sylow's theorem, there exists a Sylow $11$-subgroup of G. By Third Sylow's ...
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2answers
60 views

Prove or disprove $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$

I need to prove or disprove: To the group $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$ My attempt: I just wrote all the details that I know: element in $A_5$ should be in form like ...
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15 views

How to transform roots/weights from the simple root basis to the H-basis?

Often the roots and weights of some Lie algebra are written in terms of the simple root basis $$ r =(a_1,a_2,a_3,\ldots)=a_1 \alpha_1 + a_2 \alpha_2 + a_3 \alpha_3 +\ldots,$$ where $α_i$ denotes the ...
4
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1answer
28 views

Simple Roots of E6 in Coordinates?

There are several possibilities how one can write simple roots in terms of coordinates. Firstly, they depend on the numbering of the nodes in the Dynkin diagram. Let's fix the choice for $E_6$ to be ...
11
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4answers
715 views

Product of all elements in finite group

Question: If $G$ is a finite group such that the product of its elements (each chosen only once) is always $1$, independent of the ordering in the product, what can we say about $G$? I was trying to ...
2
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43 views

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $ which has a complement $ X $ in $ Q $. If $ Q = \langle y \rangle X $ for some element $ y $, then ...
3
votes
2answers
87 views

Finding order of $gag^{-1}$ in $G$ if $a^2=e\in G$

Let $G$ be a group, the order of $G$ is even, let $a \in G$, $a^2=e$ I need to find the order of $gag^{-1}$ in $G$ My attempt: ...