Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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1answer
233 views

If a $p$-group acts on another $p$-group by automorphisms, there is a nontrivial fixed point.

If I have nontrivial $p$-groups $G, H$, where $p$ is a prime number, and $H$ acts on $G$ by automorphisms, how can I show that the set of fixed points of the action (the set $\{ x \in G : h*x = ...
1
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1answer
124 views

The center of $A_4\times\mathbb Z_2$

Here is a simple question but I am trapped in solving the final part of it: Show that $Z(A_4\times\mathbb Z_2)$ is characteristic subgroup of $A_4\times\mathbb Z_2$ but not a fully invariant ...
4
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1answer
158 views

Smallest pure subgroup containing a fixed subgroup

I will ask a slightly more precise question then in the title. Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
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1answer
99 views

What is the galois group of $x+3$ or $(x+1)(x+2)$ ? How about $A(x)B(x)$?

As the title says I wonder what the galois group of $x+3$ is. Or even if that exists ? Since $x+3 = 0$ has only one zero/element I assume its the trivial group ? And what is the galois group of ...
4
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2answers
225 views

Gentle introduction into stability and classification theory

I am badly looking for a (very) gentle introduction into stability and classification theory answering at least some of the following questions: Why is a stable theory called "stable"? What is a ...
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1answer
33 views

Action of $D_{2n}$ on $\mathbb{P}^1$

Let $D_{n}=\langle a,b \ | a^n=b^2=abab=e\rangle$ be a dihedral group. Assume that $b$ acts on $\mathbb{P}^1$ by $z\mapsto \overline{z}$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$. ...
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3answers
285 views

Splitting exact sequences of finite abelian groups

I would like to find a condition for an exact sequence of abelian groups $$ 0\to H\to G\to K\to 0 $$ to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
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1answer
39 views

Holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$

I am looking for a holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$. Is it true that there is a unique effective action given by $$ a:z\rightarrow -z, \ \ \ \ b:z\mapsto 1/z $$ up to ...
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1answer
670 views

Algorithm to find conjugacy classes of subgroups/elements (in matrix groups)?

I'm looking for a simple (=doable to implement by myself) algorithm to compute the conjugacy classes of elements and subgroups of a given subgroup of $\text{P}{\Gamma}\text{L}(n,q)$. So given a group ...
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3answers
150 views

Normal subgroups of order 12 in $S_3\times S_3$

What are the normal subgroups of order $12$ in $S_{3} \times S_{3}$? I know that all the subgroups of order $12$ in $S_{3} \times S_{3}$ are isomorphic to the dihedral group of order $12$.
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2answers
84 views

For every $g\in G$ there exists an $h\in G$ such that $g = h^3$

Let G be a finite group whose order is not divisible by $3$. Show that for every $g\in G$ there exists an $h\in G$ such that $g = h^3$.
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4answers
3k views

Automorphism group of the quaternion group

Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ ...
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2answers
194 views

A presentation for $\mathbb Q_8\rtimes_{\phi}\mathbb Z_3$?

We know one of the presentation of $\mathbb Q_8$ is: $$\mathbb Q_8=\langle a,b,c|ab=c,bc=a,ca=b\rangle$$ and if we want to construct the semi-direct product of $\mathbb Q_8\rtimes\mathbb Z_3$; this ...
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3answers
201 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
21
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2answers
537 views

There are at most two prime numbers dividing $|G|$

Need just hints Let $G$ is a finite non-abelian group such that all its proper subgroups are abelian. Then there are at most two different prime numbers dividing $|G|$. I found some ideas about ...
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2answers
208 views

Show that the group $G$ is of order $12$

I am studying some exercises about semi-direct product and facing this solved one: Show that the order of group $G=\langle a,b| a^6=1,a^3=b^2,aba=b\rangle$ is $12$. Our aim is to show that ...
3
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1answer
162 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
3
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4answers
500 views

The product of all elements in $G$ cannot belong to $H$

Let $G$ be a finite group and $H\leq G$ be a subgroup of order odd such that $[G:H]=2$. Therefore the product of all elements in $G$ cannot belong to $H$. I assume $|H|=m$ so $|G|=2m$. Since ...
7
votes
2answers
752 views

Every normal subgroup of a finite group is contained in some composition series

In this context composition series means the same thing as defined here. As the title says given a finite group $G$ and $H \unlhd G$ I would like to show there is a composition series containing $H.$ ...
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0answers
87 views

Fixed points of coset operation

Let $G$ be a finite group which operates on two finite sets $E_1$ and $E_2$. Say that $E_1$ and $E_2$ are weakly $G$-isomorphic if for every $g \in G$, $\mathrm{Card}(E_1^g)=\mathrm{Card}(E_2^g)$, ...
6
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1answer
167 views

$ K(G)=3 \Longrightarrow G\cong\mathbb Z_3\ \mathrm{or} \ G\cong S_3$

According to J.S. Rose book "A Course on Group Theory": In class equation $$|G|=\sum_{i=1}^k|G:C_G(x_i)|$$ where $x_1,x_2,...,x_k\in G$ one from each of above $k$ classes; $K(G)$ is called the ...
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0answers
166 views

The first cohomology of group

I would like to ask if G is a group of order $p^4 (p\neq 2)$ as form $C_{p^3}\rtimes C_p$ (a semidirect product of cyclic group of order $p^3$ by a group of order $p$). Then can we obtain the first ...
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1answer
231 views

$p$-Sylow in quotient groups

Prove that if $P$ is a $p$-Sylow of $G$ and $N \triangleleft\> G$ then: $ PN/N $ is q $p$-Sylow of $G/N$ $P \cap N $ is $p$-Sylow of $N$
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1answer
432 views

group order $p^2q^2$ will be abelian

let $(G,*)$ a group order $p^2q^2$ such that $q\nmid p^2 -1 $ y $p\nmid q^2 -1$ then $G$ is abelian. for Sylow theorem $n_p\equiv 1\mod (p)$ then $n_p = 1, q, q^2 $ but $n_p\neq p^2$ the same form ...
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1answer
139 views

Do all representations of finite groups have one-dimensional subrepresentations?

Let V be a representation of a finite group G, and $v\in V$ - a nonzero vector. Put $$u = \sum_{g\in G} gv.$$ Then for any $g\in G$ we have $gu = u$ and therefore $<u>$ is a subrepresentation of ...
5
votes
2answers
170 views

$p$ is prime and $p^2\large\mid\normalsize|G|$

Hints needed: Let $p$ be a prime and $G$ a finite group such that $p^2\large\mid\normalsize|G|$ then $p\large\mid\normalsize|\text{Aut}(G)|$.
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2answers
340 views

A non-abelian $p$-group $G$

There is some facts about finite non abelian $p$-groups over the site. For example, when $n=3$: Nonabelian groups of order $p^3$. I have found the following problem in my very old works unsolved, ...
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2answers
938 views

When does an abelian group have a composition series?

There is an exercise in the book "An Introduction to the group theory by J.J. Rose" which can also be found as a proposition in "Abstract algebra by T. Hungerford": Every finite group has a ...
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2answers
269 views

Some irreducible character separates elements in different conjugacy classes

Let $x$ and $y$ be elements that are not conjugate in $G$. Then there is some irreducible character $\chi$ such that $\chi(x) \not = \chi(y)$. Clearly the "irreducible" part isn't important, ...
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1answer
652 views

What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
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1answer
250 views

Semidirect products of an elementary abelian p-groups and cyclic groups of prime order

(1) If $A$ is a elementary abelian p-group. And $Q=\langle t\rangle$ is a group of order q ($q\neq p$ prime numbers). For which primes p,q does the semidirect product $A\rtimes Q$ exist (so ...
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0answers
138 views

Automorphisms of a group and cyclic subgroups

I have the following question: Let $G$ be a group. And $A,B\leq G$ two cyclic subgroups with $|A|=|B|=n$. When does an Automorphism $\alpha\in\mathrm{Aut}(G)$ exists such that $\alpha(A)=B$? If ...
2
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1answer
212 views

conjugacy classes in representation theory

I have a question on conjugacy classes in this post, especially to this sentence: "if $g$ is a rotation of order $5$, then it is $K$-conjugate to each of $g^{13},g^{13^2},g^{13^3},g^{13^4}=g$". ...
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1answer
339 views

program to find matrix group given generators (Matlab)?

Given a small number of generators for a group of small (e.g. $3\times3$) matrices over a finite field $\mathbb{F}_p$ where $p$ is prime, how can we find all the elements of the group? The best I've ...
1
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1answer
167 views

$H\trianglelefteq G$ is maximal then $H=C_G(H)$

Let $p$ is a prime and $G$ is a finite $p$-group. Also, the normal subgroup $H$ of $G$ is maximal among abelian subgroups of $G$ which are normal in $G$ as well. Prove $H=C_G(H)$. I should ...
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3answers
87 views

How to expand a representation

If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there ...
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4answers
225 views

$p\mid [G:H]$ then $p\mid [N_G(H):H]$

I encountered the following problem for the first time. I sketched a proof for it. I will be thankful if I know it is correct or not. Thanks. $p$ is a prime and $H$ is a $p$-subgroup of a finite ...
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2answers
65 views

$H\vartriangleleft G$ and $|H|\not\equiv 1 (\mathrm{mod} \ p)$ then $H\cap C_{G}(P)\neq1$

Let $G$, a finite group, has $H$ as a proper normal subgroup and let $P$ be an arbitrary $p$-subgroup of $G$ ($p$ is a prime). Then $$|H|\not\equiv 1 (\mathrm{mod} \ p)\Longrightarrow H\cap ...
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0answers
78 views

Connection between: Expand a representation and dual representation

I worked the proof for 3) in this link out, but I have problems with the last step: Let $\sigma$ be a irred. representation of a normal subgroup $H=\langle z\rangle$ of $G$ and $\sigma'$ its dual ...
2
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1answer
269 views

Number of elements of order $7$ in a group of order $28$

Given a group $G$ with order $28 = 2^2 \cdot 7$. Sylow-Theory implies that there is a exactly one $7$-Sylow-Subgroup of order $7$ in $G$, and $1$ or $7$; $2$-Sylow-Subgroups. Where to go from here ...
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2answers
309 views

$G/H$ is a finite group so $G\cong\mathbb Z$

Let $G$ is an abelian infinte group such that for all nontrivial subgroups $H$ $$\forall H\leq G, \left|\frac{G}{H}\right|<\infty$$ Prove that $G\cong\mathbb Z$. What I have done: Clearly, it ...
8
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2answers
369 views

What is an irrreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
2
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1answer
182 views

Question of Clifford theory

I have some questions about thist post: faithful irreducible representations of cyclic and dihedral groups over finite fields I would appreciate it really if someone could help me. 1) Do I get with ...
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4answers
256 views

$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.

I've tried, but I can't solve the question. Please help me prove that: $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
3
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1answer
253 views

Why are all groups of order 153 abelian?

$153 = 3^2 \cdot 17$ so lets assume there are $s_3$ $3$-Sylow-Subgroups and $s_{17}$ $17$-Sylow-Subgroups. We know that $s_3 \mid 153$ so $s_3 \in \{1,3,9,17,51,153\}$ and $s_{17} \in \{1,17,51\}$. ...
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2answers
728 views

Non-abelian group $G$ of order $p^3$

I just need some hints to prove this: Let $|G|=p^3$ be a a non-abelian group. If every subgroup of $G$ is normal, then $p=2$ and $G=Q_8$. I know the following facts about a non-abelian group ...
5
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1answer
118 views

Is this problem correct that $HG'=G$?

Here, I have the following homework: Let $G$ is a finite $p-$group and let $H$ be a subgroup of it such that $HG'=G$. Prove that $H=G$ ($G'$ is the commutator subgroup). I have tried to show ...
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2answers
311 views

Converting GAP groups into SAGE permutation groups.

I have been working with SAGE online, and have made some programs to test some hypothesis about finite groups. However, the pre-defined "named" groups in SAGE are quite limited (basically, the ...
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4answers
88 views

$(p,\frac{n}{p^\alpha})=1$ then $p\nmid\binom{n}{p^\alpha}$

Let $n=p^\alpha m$ wherein $(p,m)=1$. Then we have $$p\nmid\binom{n}{p^\alpha}$$ What I have done is just playing with $\binom{n}{p^\alpha}$ ...
3
votes
2answers
303 views

An explicit calculation of Galois group

This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...