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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Generalisation of automorphisms on groups of units $U(p^n)$?

I tried to solve the following exercise: Prove that the mapping from $U(16)$ to itself given by $x \mapsto x^3$ is an automorphism. What about $x \mapsto x^5$ and $x \mapsto x^7$? Generalize. ...
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1answer
62 views

More question on the proof of orbit-stabilizer theorem from Gowers's weblog

Still I'm reading Gowers's weblog about orbit-stabilizer theorem, I must admit that my understanding of this materiel improved, but still I have some question. Let $G$ be a finite group, and $X$ be ...
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60 views

The centralizer of an involution in $PSL(2, q)$ is a dihedral group of order $q \pm 1$.

Let $G \cong PSL(2, q)$ with $q \equiv 3 \pmod{8}$ or $q \equiv 5 \pmod{8}$. If $u \in G$ is an involution, then $C_G(u)$ is a dihedral subgroup of order $q \pm 1$. I know that $PSL(2,q)$ could be ...
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2answers
34 views

What does “the subgroups of $G$ form a chain” mean?

I am being asked to show that: $G$ is a cyclic $p$-group $\iff$ its subgroups form a chain. What does "its subgroups form a chain" mean? Please keep in mind that I am just asking for the meaning of ...
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1answer
31 views

Let $m$ be the number of solutions of the equation $x^2 = 1$ in the ring $Z_n$, where $Φ_n$ is Euler group, $n> 2$.

Let $m$ be the number of solutions of the equation $x^2 = 1$ in the ring $Z_n$, where $Φ_n$ is Euler group, $n> 2$. Prove that $m$ is an even number. Could anyone help me or give me a hint? ...
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1answer
39 views

Number of $p$-groups of small order and of exponent $p$

In a very recent paper of MR Vaughan-Lee, it is proved that the number of $p$-groups of order $p^8$ and exponent $p$ is a polynomial (of fourth degree) in $p$. Let us consider $p$-groups of order $&...
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1answer
31 views

Restriction of automorphism to the generalized Fitting subgroup

for my master's thesis I am going through M.Hertweck and W.Kimmerle's article on Coleman automorphisms. I encountered the following reasoning, which I can't seem to follow. We know the following on ...
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1answer
35 views

Let $0 \to \mathbb{Q} \xrightarrow{f} \mathbb{Q}^{m} \to \mathbb{Q} \to 0$ be a exact sequence is $m=2$ always?

Let $0 \to \mathbb{Q} \xrightarrow{f} \mathbb{Q}^{m} \xrightarrow{g} \mathbb{Q} \to 0$ be a exact sequence, since $f$ is injective it's clear that the image of $f$ is isomorphic to $\mathbb{Q}$. So ...
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1answer
37 views

Explain the order of elements in $\mathbb Z_{12}$

I cannot understand the cardinality of elements in modular classes like here explained, source of the latter is C1080. Definitions $\mathbb Z_{12}=\{\overline 0,\overline 1,\ldots, \overline{11}\}...
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1answer
21 views

Determine which abelian groups can be the central term of this exact short sequence

I am trying to solve the following problem: Determine which abelian groups $A$ can appear as central terms in a short exact sequence $\mathbb{Z} \to A \to \mathbb{Z} \oplus \mathbb{Z}_5$ What I'...
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1answer
23 views

Prove that $AN$ is a subgroup of $G$ if $A$ and $N$ are its subgroups and $N$ is normal in $G$

Let $A$ be a subgroup of a group $G$, and let $N$ be a normal subgroup of $G$ Prove than in this case $AN = \{an|a \in A, n \in N \}$ is a subgroup of $G$ I know that since $N$ is normal, we have $AN ...
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1answer
10 views

Showing that functional composition of a function is close

For a fixed point $\big(a,b\big)$ in$ R^{2}$ define $T_{a,b} :R^{2} \rightarrow R^{2} by \big(x,y\big) \rightarrow \big(x+a,y+b\big).$ then $ G= \big\{T_{a,b}| a,b \in R\big\}$ is a group ...
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2answers
51 views

Finding elements of a set that is itself a group under addition.

Let p and q be distinct primes. Suppose that H is a proper subset of the integers and H is a group under addition that contains exactly three elements of the set $\big\{p,p+q,pq,p^{q},q^{p}\big\} .$...
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1answer
48 views

Frattini Subgroup of Free Group

It is perhaps around 1950, proved by Higman and Neumann that the Frattini subgroup of a Free group is trivial. I want to know, now, is there elementary proof of this? If yes, please write short ...
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1answer
42 views

Translating an equation from multiplicative to additive form

The equation I have is $ \big(ab^{2}\big) ^{-3}c^{2}=***e***$ which is in multiplicative form but which I am trying to convert it to additive form. e is the identity element Here's what I have so ...
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36 views

On the automorphism group of the dihedral group $\text{Aut} (D_3)$

I am trying to calculate $\text{Aut} (D_3)$, the automorphism group of the group of symmetries of the triangle. But I got stuck and now I have two questions about this. Let me share my thoughts first:...
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2answers
44 views

When is every group of order $n$ nilpotent of class $\leq c$?

In the paper 'Nilpotent Numbers' by Pakianathan and Shankar (http://www2.math.ou.edu/~shankar/papers/nil2.pdf), it was proven that every group of order $n$ is nilpotent if and only if $p^k\not\equiv 1\...
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0answers
61 views

Center of the group algebra of the symmetric group

How to prove that the center of the group algebra of the symmetric group is generated by 1-cycle conjugacy classes? I mean, that the center (consisting on class functions) is multiplicatively ...
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1answer
51 views

$\pi$ -Hall normal subgroup is characteristic

I've this exercise on my textbook: "Show that if G is a group (not necessary soluble), a normal $\pi$ -Hall subgroup is characteristic." I've tried to resolve it in the following way. Let $\alpha$...
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1answer
38 views

Normal Subgroup and Equal Cosets Proof

I started learning group theory and got stuck with one proof... A normal subgroup $N \subseteq G$ is defined as: $\forall g \in G; n \in N \space\ (g*n*g^{-1} \in N) $ How can I show that, using ...
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56 views

Showing $\Bbb Q_8$ is nilpotent.

My definition of nilpotent is Definition: A group $G$ is said to be nilpotent if $C_{i}(G)=G$ for some $i$. Where $C_i(G)=\tau^{-1}(Z(G/C_{i-1}(G)))$ $$\tau:G\to G/C_{i-1}(G), \text{ by } \tau(a)=...
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72 views

Order of an Equivalence Class and the number of Coset.

Let $ \mathcal{M} $ be the set of all subsets of finite group $G$ which have $p^{\alpha} $elements. Thus $ \mathcal{M} $ has $ {p^{\alpha}m \choose p^{\alpha}} $ elements. Given $M_1 ,M_2 \in ...
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0answers
17 views

A group with unusual discrete log properties.

Does there exist a group where computing $g^x$ from $g^{a^{x}}$ is easy, computing $g^{a^{x}}$ from $x$ and $g^{a}$ is hard, and computing $x$ from $g^a$ and $g^{a^x}$ is hard. Intuitively I would ...
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1answer
59 views

Show that $|C_G(u)| = 12$ by “counting involutions”

Let $G$ be a transitive permutation group acting on $\Omega$ such that every non-trivial element fixing some point has exactly three fixed points. Suppose $G_{\alpha} \cong A_5$ for some point ...
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3answers
82 views

Multiplicative group $(\mathbb R^*, ×)$ is group but $(\mathbb R, ×)$ is not group, why?

$\mathbb R^*$ refers to $\mathbb R$ without zero. Please explain the statement "$\mathbb R$ is not a group under multiplication, it is a group under addition." in the comment. Basically: Why is ...
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2answers
50 views

Finite Fields problem [closed]

"Given a Galois Field $(\mathbb{F}, +, \cdot)$ of order 8. With an element $x \in \mathbb{F}$ we create a group $(\{x^m | m \in \mathbb{Z}\}, \cdot)$. ($x^m$ is calculated via the second operator $\...
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35 views

Meaning of abelian subquotient

I was reading an article, somewhere it says that the "abelian subquotients of the group $G$" are .... How does it defined ? For example if we take $G=S_n$, the symmetric group, then what are the ...
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3answers
86 views

Explain $\langle \emptyset \rangle=\{1\},\langle 1 \rangle=\{1\}. H\leq G \implies \langle H\rangle=H.$

On page $61$ of the book Algebra by Tauno Metsänkylä, Marjatta Näätänen, it states $\langle \emptyset \rangle =\{1\},\langle 1 \rangle =\{1\}. H\leq G \implies \langle H \rangle =H$ where $H \...
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1answer
26 views

If $G = VN$, $V$ a four group and $N$ regular normal, then there exists some Sylow subgroup left invariant by $V$

Let $G$ be a permutation group on $\Omega$ with $G = VN$, where $V \cong C_2 \times C_2$ (the four-group) and $N$ has odd order with some prime divisor $>3$. Suppose $N$ is a regular normal ...
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53 views

Showing that the right and left regular representations are equivalent

Let $G$ be a finite group. Let $C(G,\mathbb{C})$ be the complex vector space of all functions from $G$ to $\mathbb{C}$. We define two representations of G on $C(G,\mathbb{C})$: the left regular ...
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1answer
24 views

Verifying if a general linear set is a group

This is an example from my text: Let $\textit{F}$ be any of $Q,R,C,Z$ or$ Z_{p}\left (p is a prime. \right)$ The set $GL\left(2,\textit{F} \right)$ of all 2 x 2 matrices with nonzero determinants ...
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1answer
30 views

how to find out coset representation of the following subgroup of the given group?

$G$ is a group of $2\times 2$ matrices= $SL(2,\mathbb{Z})/\{I_2,-I_2\}$ where $SL(2,\mathbb{Z})$ is invertible matrix with entries in $\mathbb{Z}$ and determinant $1$. then I know that G is generated ...
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5answers
23 views

Elements in composition of groups.

Say $|A|=|B|=k$ and $A\cap B = id$ and $A,B$ are subgroups of $G$. $AB$ is then also a subgroup of $G$ provided that $A,B$ commute. But what is $|AB|$? Is it simply $k^2$?
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24 views

Elementary group theory: counting the order of cosets.

If $H\le G$ and $|H|= n$ then what is $|xHx^{-1}|$ where $x\in G$? Is it simply $n$, because if $x\in H$ then $|xHx^{-1}|=|H|=n$ and if $x\notin H$ then clearly $|xHx^{-1}|=n$ as well. Is this ...
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2answers
40 views

Why must $A_n$ be generated by the 3-cycles

For my course in Group Theory, I have seen various proofs that show why the alternating group $A_n$, which consists of the elements of $S_n$ that can be expressed as an even number of transpositions (...
2
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2answers
65 views

$(G,+)$ abelian group is divisible $\Longleftrightarrow$ it's an homomorphic image of $\Bbb Q^{(X)}$

Let $(G,+)$ be an additive abelian group. Let us suppose $G$ divisible (i.e. $G=nG\;\;\;\forall n\ge1$). Let then $x,y\in G$. Then there exists $z\in G$ and $n,m\ge1$ such that $x=nz$ and $y=...
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1answer
48 views

Question on the Orbit-Stabilizer theorem

I'm trying to get a deeper understanding on Orbit-Stabilizer theorem and I came across with gowers excellent post explaining the intuition behind the theorem. I will quote two statements from there, ...
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1answer
30 views

Prove that $G/N = \{1N\}$ iff $G=N$

Prove that $G/N = \{1N\}$ iff $G=N$ Where $N$ is a normal subgroup of $G$, a finite group. I'm having trouble proving this. Could someone please help me? Thank you.
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53 views

Conjugates of a sylow 2-subgroup of $A_5$

The exact problem is as follows: Show that a Sylow 2-subgroup of $A_5$ has exactly 5 conjugates. My question is why are there not 15 conjugates. Seeing as every Sylow 2-subgroup of $A_5$ is the ...
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1answer
28 views

Explanations on the proof of lemma 2.5 of hungerford's Algebra?

Lemma 2.5.(v) $p^m\Bbb Z_{p^n}\cong \Bbb Z_{p^{n-m}}\ (m<n)$ where $p$ is prime. And the proof goes $\ldots$ Note that $p^m\in \Bbb Z_{p^n}$ has order $p^{n-m}$. Therefore $p^m\Bbb Z_{p^n}=\...
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2answers
42 views

A subtle error in a “change of variable” in a sum

Let $G$ be a finite group of order $n$, and let $E$ be a finite set. Let $\star$ be an action of $G$ on $E$. Suppose that $G \star x_1,..., G \star x_m$ are the distinct orbits of elements in $E$. ...
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0answers
126 views

Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
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1answer
44 views

If $G_{\alpha} \cong S_4$ and $|\mbox{fix}(g)| \in \{0,3\}$ for $g \ne 1$. Then $G$ has transitive normal subgroup of index $2$.

Let $G$ be a transitive permutation group such that $|\mbox{fix}(g)| \in \{0,3\}$ for every nontrivial $g \in G$. Also suppose $|N_G(G_{\alpha}) : G_{\alpha}| = 1$, i.e. $G_{\alpha}$ is the only fixed ...
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1answer
41 views

Surjective Homomorphisms of Isomorphic Abelian Groups [duplicate]

Is a surjective homomorphism between two (abstractly) isomorphic finitely generated abelian groups necessarily an isomorphism? I know this is true if the groups are torsion (finite) or torsion-free. ...
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2answers
88 views

can anything be (re)defined in math? [closed]

anything in particular this case is referring to algebra what I mean is this: I am looking at group theory right and the professor says something along the lines of "the set {-1,+1}, under *, defined ...
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0answers
27 views

Is $C_{S_n}(g) = H \times S_m$?

Let $g$ be an element of the symmetric group $S_n$. If $c$ commutes with $g$, then $c$ permutes the set of $g$-fixed numbers in $\{1,\ldots,n\}$. Write $C_{S_n}(g)$ for the centralizer of $g$ and ...
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2answers
80 views

If $G = \{a + b: a, b \in A\}$ a group, then is $A$ a group?

Let $A $ be a subset of an abelian group $H$. Then if $G = \{ a + b : a, b \in A\}$ is a group and $A$ is closed under taking negatives, then is $A$ also a group?
1
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1answer
49 views

Let $N$ be maximal normal subgroup with $TU \cap N = 1$, where $TU = N_G(U)$. Then $N$ is nilpotent and contains certain involutions.

Let $G$ be a finite group with subgroup $U \le G$ of odd order such that $U\ne U^g$ implies $U\cap U^g = 1$. Suppose there exists an involution $t \notin U$ such that $N_G(U) = TU$ with $T = \langle t ...
1
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0answers
31 views

Proving that a finite group $G$ is solvable iff for every divisor $n$ of $|G|$ such that $(n, |G|/n) = 1$, $G$ has a subgroup of order $n$.

I found this theorem in Dummit and Foote, and there was no proof of it there. It looks difficult to prove and I also could not find any resources online to help me out with this theorem. So here is ...
2
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1answer
41 views

Significance of the notion of equivalent actions vs. permutation isomorphic action

Let $G$ be a group acting on $\Delta$, and $H$ be a group acting on $\Gamma$. If there exists an isomorphism $\varphi : G \to H$ and a bijection $\psi : \Delta \to \Gamma$ such that $$ \psi( \alpha^g ...