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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Subgroups which contain all $p$-Sylowsubgroups for some fixed prime $p$

Is it true that if some subgroup $H \le G$, $H\ne G$ contains all $p$-Sylowsubgroups for some fixed prime $p$, then $H$ contains some non-trivial normal subgroup of $G$?
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Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$

For an arbitrary group $(G,\cdot)$ let $\operatorname{Aut}(G) = \{f: G \to G \mid f \text{ is an isomorphism}\}$ be the set of all automorphisms of the group $G$. We assume that ...
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What does it mean to be a real Lie group

What does it mean to be a real Lie-group ? For example it is said that $SU(N)$ is a real Lie-group. While for example for $SU(2)$ the 2 dimensional matrix-representation consists of the Pauli ...
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Nonempty, associative, and closed under inverses but not a group

Given an example of a set $G$ and an operation $*$ on $G$ such that $*$ is not a binary operation on $G$ but associative, identity and inverses properties hold? Basically, try to find an example to ...
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$P\in Syl_p(G)\Rightarrow|G:N_G(P)|=|Syl_p(G)|$

Let $G$ be a finite group s.t. $|G|=p^rm$, where $(p,m)=1$. Let then $P$ be a $p$-Sylow subgroup of $G$, i.e. $P\le G$ with $|P|=p^r$. We want to show that $|G:N_G(P)|$ is the number of $p$-Sylow ...
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Prove $G$ is a group under $\ast$ given as $(a, b)*(c, d) = (ac,bc+d)$

Let $G = \{(a, b) \text{ }|\text{ } a, b \in \mathbb{R}, a \neq 0 \}$. Let $*$ be an operation on $G$ defined by $$(a, b)*(c, d) = (ac,bc+d)$$ Prove $G$ is a group. Is this a group or not? because ...
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What can we say about this quantity?

Let $\phi(n)$ be the Euler phi-function. If $a>1$ is an integer, then what is the remainder when $\phi(a^n - 1)$ is divided by $n$ in accordance with the Euclidean algorithm?
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Automorphism groups of isomorphic groups are isomorphic

Say $G \cong H$ are isomorphic groups. Show $Aut(G) \cong Aut(H)$ I just made this up so I'm not sure if actually $Aut(G) \cong Aut(H)$ is true but I'm $99.9\%$ sure this should be true I'm having ...
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99 views

$G=\bigcup\limits_{i=1}^ \infty G_i$ and $G_n \subset G_{n+1} , \forall n \in \mathbb N$ $\implies$ $G$ is not cyclic

Let $G$ be a group such that $G=\bigcup\limits_{i=1}^ \infty G_i$ and $G_n \subset G_{n+1} , \forall n \in \mathbb N$ , then how does one prove that $G$ is not cyclic ? Each $G_i$ is a subgroup of ...
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100 views

Is this subgroup normal?

Let $T$ be a cyclic subgroup of a group $G$ such that $T$ is normal in $G$. Let $S$ be a subgroup of $T$. What can we say about whether or not $S$ is normal in $G$? My work: Let $T \colon = ...
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29 views

Raising multiplied group elements to a power

If for example we have $(aba^{-1})^n$, how do you go about expanding this to show that it's the same as $a^nba^{-n}=ab^na^{-1}$?
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89 views

cyclic subgroup elements

I'm having hard time finding elements of the cyclic subgroup $\langle a\rangle$ in $S_{10}$, where $a = (1\ 3\ 8\ 2\ 5\ 10)(4\ 7\ 6\ 9)$ This is my attempt: \begin{align} a^2 &= (1\ 8\ 5\ 10)(4\ ...
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First isomorphim theorem $\phi(H) = HN/N$

Theorem: $\frac{H}{H \cap N} \simeq \frac{HN}{N}$. Define $\phi : H\rightarrow G/N$ by $\phi(h)=hN \ \ \ \forall h \in H$ I'm not sure why the image, $\phi(H) = HN/N$ Can anyone show me why? My book ...
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Finding another representation of a group represented by a set of matrices

I want to find the representation of the group $$ \left\{ \left. \left( \begin{array}{cccc} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ c & 0 & ad & 0 \\ 0 & ...
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Non-commutative quotient group?

If you have a non-abelian group $G$ with some normal subgroup $K$, is it possible to have a non-abelian quotient group $G/K$? Besides actually sitting down and trying to generate quotient groups ...
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$\mathbb{Z}G$ is necessarily a subgroup of $G$, or am I missing something?

Let $\mathbb{Z}G=\left\{\sum\limits_{g\in G}n_gg\mid n_g\in\mathbb{Z},g\in G\right\}$, where if $G$ is infinite, we only consider finite formal sums of elements of $G$ with coefficients in ...
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114 views

the nilpotency class of Frobenius kernel

As we know, if G has a fixed-point-free automorphism of order p, then G is nilpotent, can we know something about the nilpotency class of Frobenius kernel ?
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262 views

Commutator Subgroup is Normal Subgroup of Kernel of Homomorphism

Please help to understand this problem. Let $G$ be a group, $H$ an abelian group, $\phi : G \rightarrow H$ a homomorphism. Show that $C(G) \lhd \mathrm{Ker}(\phi)$ I must be misunderstanding ...
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106 views

Order of conjugate of an element given the order of this element

Let $G$ is a group and $a, b \in G$. If $a$ has order $6$, then the order of $bab^{-1}$ is... How to find this answer? Sorry for my bad question, but I need this for my study.
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why $\mathbb{Z}_8$, $\mathbb{Z}_4\times \mathbb{Z}_2$ and $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ are not isomorphic groups.

And find an example of a group of order 8 that is not isomorphic to $\mathbb{Z}_8$ or $\mathbb{Z}_4\times \mathbb{Z}_2$ or $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ Thank you.
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For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
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123 views

There exist two distinct elements of $G\setminus H$ which commute?

Let $(G, *)$ be a finite group of odd Because $G$ has odd order, every element x frhas order greater than 2, so I think, the condition that H is non-commutative is just to confuse.
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117 views

Nonabelian group of order $p^4$ [closed]

Let $P$ be a nonabelian group of order $p^4$, where $p$ is a prime, and let $A$ be a subgroup of $P$ maximal with the property of being normal and abelian. Prove that $A$ is of order $p^3$. Thanks a ...
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93 views

Exact sequences and (semi) direct product

Let $0 \longrightarrow H \longrightarrow G \longrightarrow K \longrightarrow 0$ be a short exact sequence. My question is quite general: how can one recover $G$ knowing $H$ and $K$? Naive question: ...
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122 views

prove that the Galois group $Gal(L:K)$ is cyclic

Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is ...
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369 views

Non-trivial nilpotent group has non-trivial center

A book I'm reading quotes the following result without any explanation: Any non-trivial nilpotent group has a non-trivial center. (The definition of "nilpotent group" is as follows: Suppose $G$ is a ...
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I need to prove that $\mathbb{C}^*/\mathbb{R}^+\cong U$

I need to prove that $\mathbb{C}^*/\mathbb{R}^+\cong U$ by the theorem: $G/\ker(\varphi)\cong Im(\varphi)$. $U$ - Is the circle unit. $\mathbb{R}^+=\left\{x\in \mathbb{R}\Big|x>0\right\}$ (So the ...
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Every abelian group can be embedded in a divisible group

I've a proof of the title-statement if I can prove the following: every cyclic group can be embedded in $\mathbb{C}^{\star}$, the multiplicative group of complex numbers. Can you suggest me how to ...
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177 views

Groups of prime squared order

I have done part $a$ and $b$, stuck on $c$. I don't know what to do here. I know that since $G$ has order $p^2$, any element must have order $1, p$ or $p^2$ so any generator must have one of those ...
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Prove that D is normal

Let G be a group, $T = G \times G$ and let $D = \{(g,g)\in G \times G | g\in G\}$. Prove that D is normal in T if and only if G is abelian. I assume that D is normal in T, then for any $x,y \in T$ ...
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How to multiply permutations

I didn't understand the rules of multiplying permutations. I'll be glad if you can explain me... For example: we have $f=(135)(27),g=(1254)(68)$. How do I calculate $f\cdot g$?? Thank you!
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Size of a subgroup generated by two elements

Suppose I have a finite group with elements $x,y$ such that $\langle x \rangle \cap \langle y \rangle = \{1\}$. Is it true that $|\langle x,y \rangle|=|\langle x \rangle|\cdot|\langle y\rangle|$? I ...
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Confused with a statement

A corollary at page 91 of the book Group Theory I by M. Suzuki is as follows: Let $A$ be an abelian subgroup of a $p$-group $G$. If $A$ is maximal among abelian normal subgroups of $G$, then $A$ ...
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Finite group of two generators

My question is simple : Any finite group of two generators is cyclic, semidirect sum, or direct sum ?
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On the definition of product of groups

What I'm asking comes from Bosch, Algebra, the first chapter on elementary group theory. 1) Let $X$ be a set and $G$ a group. Then the set $G^X$ of maps from $X$ to $G$ is a group in a natural way, ...
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311 views

Infinite coproduct of abelian groups

One can see on every text (book, lesson, comments) that a direct sum/coproduct of abelian groups is the same as a finite product but in the infinite case, the direct sum/coproduct is only a subgroup ...
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164 views

What is a good way to think of Factor Groups?

I'm having a hard time thinking about factor groups. I just don't understand what notation like $\mathbb{Z}_{60}/\langle 12 \rangle$ means. Furthermore, when asked about giving the order $26 + \langle ...
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241 views

For any finite group, there is a homomorphism whose image is simple

This is for homework. The question asks "Show that, for any finite group $G$, there is a homomorphism $f$ such that $f(G)$ is simple." My thought was this. Since $G$ is finite, there are only a ...
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90 views

Isomorphism question about groups

I know that the group $\mathbb{Z}/9\mathbb{Z}$ is not isomorphic to the group $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$, but I just do not know how to prove this.
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Subgroup of GL(2,Z) generated by two matrices.

What is the subgroup of $GL_2(Z)$ generated by the matrices: $\left( \begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \right) $ and $\left( \begin{smallmatrix} 0&1\\ 1&0 ...
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Proving that in an infinite cyclic group order of every element is infinite

Prove that in an infinite cyclic group order of every element ($\ne e$) is infinite. I have tried proving this way : If there exists an element of finite order, then it must generate a finite ...
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Power automorphism and abelian groups

First of all I'm not sure if "Power automorphism" is the correct term, so I apologize if it is not. "Let $G$ be an abelian group of order $n$, and $m$ an integer. $f:G\rightarrow G$ s.t. $f(a)=a^m$. ...
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460 views

Exponent of a finite abelian group

I have a very basic question: Let $G$ be a finite abelian group and let $m$ be the exponent of $G$. Then does there exist $g\in G$ s.t. o$(g)=m$ and if so, why? Many thanks in advance.
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232 views

Non-abelian groups where $(ab)^{n+i}=a^{n+i}b^{n+i}$ for all $0\leq i\leq k$, $k>1$

In this question it is asked for an example of a group $G$ such that $(ab)^n=a^nb^n$ for all $a, b\in G$ holds for two consecutive integers $n\in\{m, m+1\}$, but $G$ is not an abelian group. The ...
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92 views

What powers of an $18$ cycle permutation are also $18$ cycles?

Suppose we are given a permutation $A = (1,2,...,18)$ which is an $18$ cycle in $S_{18}$. We want to find $i \in \mathbb{Z}$ such that $A^i$ is also an $18$ cycle. Now I know how to do this by trial ...
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Relationship between $PGL_2$ and $PSL_2$

I read somewhere that $PGL_2(\mathbb{C})=SL_2(\mathbb{C})/N$ where $N$ is the normal subgroup consisting of $\pm \left( \begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right)$. It is unclear to me ...
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105 views

Metanilpotent groups and saturated formation

Class of all metanilpotent groups is a saturated formation ? How do I prove
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153 views

Permutation representation of group described by $a_i^2=\theta^2=1, a_ia_{i+1}=\theta a_{i+1}a_i=a_{i+2}$.

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
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What to call $x^{-1}ax$?

If $G$ is a group and $a,x\in G$, then would we call $x^{-1}ax$ a conjugate of $x$ or a conjugate of $a$? Sorry for such a short question, was just doing a problem and want to call this something so ...
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93 views

Prove that $(S,\circ)$ is a non-abelian group

Let $S=\mathbb{R}^{*}\times\mathbb{R}$. We define a binary operation $\circ$ on $S$ by $$(u,v)\circ(x,y)=(ux,vx+y).$$ Prove that $(S,\circ)$ is a non-abelian group.