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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34 views

Why is it that if $|G|=p^{n}$ then $|Z(G)|\neq p^{n-1}$?

I am reading a proof about why If $|G|=p^{n}$ (where $p$ is prime) then $|Z(G)|\neq p^{n-1}$? That proof says that if $|Z(G)| = p^{n-1}$ then $G/Z(G)$ is cyclic which makes $G$ abelian. My question ...
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1answer
24 views

Proof for generator of the group of integer under addition modulo

Theorem: An integer $k$ in $\mathbb{Z}_{n}$ is a generator of $\mathbb{Z}_{n}$ If and Only if $gcd\left ( n,k \right )=1$ My problem lies with proving the "If" condition and here is my attempt: ...
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1answer
45 views

Using Irreducible Group Characters to Count nth Roots of Group Elements

Given $n\in\mathbb{N}$, define $\tau_n(g)=|\lbrace h\in G: h^n=g\rbrace|$. Let $\chi_i,1\leq i\leq r$ be the distinct complex irreducible characters of a finite group $G$, and let $\gamma_n(\chi_i)=\...
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33 views

Can the number of non-solvable groups of a given order be easily determined?

It can be extremely difficult to find the number of groups of a given order. But if we only want to find the number of non-solvable groups of a given order, is there an easy algorithm doing the job ?...
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1answer
25 views

Is every non-solvable group a product of a set and a subgroup?

Every solvable group is a Zappa-Szep-product. Non-solvable groups need not be a Zappa-Szep-product, $A_6$ being the smallest counter-example. However, if we define $$S:={(),(236),(456),(246),(263),(...
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25 views

Proof for generators of cyclic group [duplicate]

Theorem: Let $G=\left \langle a \right \rangle $ be a cyclic group of order n. Then $G = \left \langle a^{k} \right \rangle$ if and only if $gcd\left ( n,k \right )=1$ I've proven the "only if" ...
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1answer
55 views

Order of element in group theory

I have started taking a course on group theory and I have some confusion on few things If $G$ is a group, and $H$ is a normal subgroup, and $P$ is a prime number then Suppose $y\,H \in G/H$ is the ...
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1answer
40 views

Question about notation in group theory

If you click on the link below you will find a theorem from Daniel Gorenstein's book "Finite Groups". I am not sure what is the meaning of the i'(x). What does the ' mean? http://prnt.sc/as5413 ...
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47 views

Given $G$ finite and $L(G)=G$ where $L(G)=\{g\in G\;|\;\alpha(g)=g, \forall \alpha\in \textrm{Aut}(G)\}$ . Prove that $G=1$ or $G=\mathbb{Z}_2$

Given $G$ finite and $L(G)=G$ where $L(G)=\{g\in G\;|\;\alpha(g)=g, \forall \alpha\in \textrm{Aut}(G)\}$ Prove that if $L(G)=G$, then $G=1$ or $G=\mathbb{Z}_2$. Here I already proved that $L(G)\leq ...
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Trouble understanding Sylow's Third Theorem

The statement of Sylow's third theorem in my text goes like this, Let p be a prime and let G be a group of order $p^km$, where $p$ does not divide $m$. Then the number $n$ of Sylow $p$-subgroups of $...
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43 views

For which finite groups $G$ does a finite group $H$ exist, such that $Aut(H)$ is isomorphic to $G\ $?

Given a finite group $G$, how can I check whether a finite group $H$ exist, such that $\operatorname{Aut}(H)$ is isomorphic to $G$ ? Here http://groupprops.subwiki.org/wiki/...
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1answer
41 views

Distributive subgroups lattice

Let $G = \langle a \rangle \times \langle b \rangle $ ($a,b \in G$), where $ |\langle a \rangle| = n, |\langle b \rangle| = m$ and $gcd(n,m) = d > 1$. I need to show that subgroup lattice of $G$ is ...
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1answer
174 views

Number of $p$-groups with order $p^k$ and $|Z|=p$

What is known about the number of groups of order $p^k$ with $|Z|=p$ , which I denote $N(p^k)$ ? For $k=1$ , it is clear, that we have $N(p^k)=1$ : We have one group of order $p$ and since it is ...
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1answer
33 views

Center of a semidirect product

Here http://planetmath.org/node/87994 a formula for the center of the semidirect product of two groups for a given homomorphism is given. I also wonder whether the formula is correct or not. The ...
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1answer
59 views

A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

Let $G$ be a finite group and $H$ a subgroup. Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ ...
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12 views

Gramian of a permutation group orbit

let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure: $$ W^TW=(P_{1}t,\cdots,P_{k}t) $$ with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...
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27 views

Sylow p-subgroups of a finite group G which are contained in some other subgroup H are also Sylow p-subgroups in H

I have found a proof to the following question but I don't quite understand it. Let $P \leq H \leq G$ and $|G|=p^\alpha m$, where $p$ doesn't divide $m$. I need to show that if $P \in Syl_p(G)$, ...
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34 views

Representation of a finite group and its Sylow $p$-subgroup

Let $G$ be a finite group with order $|G|=p^n \cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
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Subgroup lattice of UT(3,3)

I need to draw a subgroup lattice of $ UT(3,\mathbb{Z_{3}}) $, the group of upper triangular matrices with diagonal one. How to do it? And whether there is somebody ready image? (I know that it can ...
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40 views

Generalization of Sylow's third theorem

Suppose $p^b||G|$, $0\le a\le b$. $H\le G$ is a subgroup of order $p^a$. Show that the number of subgroups of $G$ containing $H$ and of order $p^b$ is congruent to $1$ mod $p$. By Wielandt's proof ...
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1answer
51 views

Induction backwards to prove Sylow's first theorem

Claim: Suppose $H\le G$ and $P$ is a Sylow $p$-subgroup of $G$. Show that, without reference to Sylow's theorems, there exists some conjugate of $P$ whose intersection with $H$ is a Sylow $p$-subgroup ...
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Let $H=\{I, (12)(34), (13)(24), (14)(23)\}$. Show that $H$ is a normal subgroup of $S_4$, so that $S_4/H$ has order six. [closed]

Let $H=\{I, (12)(34), (13)(24), (14)(23)\}$. Show that $H$ is a normal subgroup of $S_4$, so that $S_4/H$ has order six. I don't see any quick way of showing that it's normal. I haven't actually ...
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1answer
24 views

Isomorphism on group - need a help. [duplicate]

Consider the following question: Let $\mathbb{Q}$ be the field of all rational numbers. Let Aut($\mathbb{Q}$) be the group of all Automorphism on $\mathbb{Q}$ (All Isomorphism from $\mathbb{Q}$ to $\...
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26 views

Some properties of finite group of order $p^aq^b$

Let $G$ be a finite group of order $p^aq^b$ ( $p$, $q$ are two distinct primes and $a, b\geq 1$) with $\operatorname{Z}(G)=1$ and $P\in \operatorname{Syl}_p(G)$, $Q\in \operatorname{Syl}_q(G)$. Also ...
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2answers
22 views

Proving this relation given two existing relations

I have the relations $a^9=b^2=1$ and $a=ba^kb$ for some $k$. I want to prove that $a^{k^2}=a$, but I'm stuck. My computation so far gives: $$a=ba^kb\Rightarrow ab=ba^k\Rightarrow a^{k-1}ab=a^{k-1}ba^k\...
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53 views

$H$ is a subgroup of $G $ with smallest possible prime index

Let $H$ be a subgroup of $G $ with smallest possible prime index. Then $H$ is normal in $G$. Above exercise is one of the classical exercises in group theory. The classical solution depends on group ...
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1answer
82 views

Sum of sum of elements in conjugacy class is a multiple of them if and only if $G=G'$

I have another question on character/group theory. This one seems to be a bit harder. Let $Cl(g_1),...Cl(g_r)$ be the conjugacy classes of a finite group, $G$ and let $C_i \in \mathbb{C}(G)$ (the ...
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24 views

Cauchy's theorem spin-off

In group theory we know from Cauchy's theorem that any finite group of order n has at least one subgroup of order p, if p|n. How can we prove the following statement: "If G is a finite group of order ...
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21 views

G is cyclic if it is a finite p-group and has only one maximal subgroup [duplicate]

How can I show that for a finite $p$-group $G$,$G$ is cyclic if it has just one non trivial maximal subgroup?
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1answer
66 views

Understanding a group of order $2^{25}.97^2$

Let $G$ be a semidirect product of a Sylow 2-subgroup $P$ and a normal subgroup $Q$. $P$ is itself is semidirect product as defined below: $$P=(\langle u \rangle \times \langle v \rangle \...
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Calculating cyclic extensions of a group

I am having trouble understanding how to calculate all possible cyclic extensions of a group. I have been following the text 'A Course In Group Theory' by John F. Humphreys, and also referring to ...
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1answer
45 views

Questions about the Zappa–Szép product

Which groups with composite order are not a Zappa–Szép product of smaller groups ? A solvable group with a composite order is always a Zappa–Szép product of smaller groups, but I am not sure about ...
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1answer
46 views

Is there a non-solvable number NOT divisible by $3\ $?

Here https://oeis.org/A056866 it is claimed that every non-solvable number is divisible by $4$ and either $3$ or $5$. However, I did not find a number in the list not divisible by $3$. So, my ...
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21 views

Levi-Civita Determinant vs Laplace Expansion

Is there anyway to derive the Laplace Expansion for the determinant from the Levi-Civita definition of the determinant? I have gotten to this point for the determinant of matrix A.
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35 views

Find some subgroups of a group with 36 elements

Let a group $G$ with $\left | G \right | = 36$, then $G$ contains a normal subgroup of order $9$, $18$ or $3$. I can see groups of order $9$ in the case $n_{3} = 1$, by Silow's Theorem, but I do not ...
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2answers
37 views

Subgroup half as big as its group is normal. [duplicate]

There is a group $G$ and subgroup $H \subset G$. Their orders: $$\left | H \right| = n, \left |G \right| = 2n. $$ How can I prove that $H$ is a normal subgroup?
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21 views

Number of ways to divide 4 points in 6 columns with constraints

While studying the Mathieu groups and more specifically the Golay code (which is closely related to the Mathieu 24-group), I encountered this paragraph in Robert Wilson's book "The Finite Simple ...
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1answer
69 views

On normal $p$-complements

This is question 5E.3. of Isaacs's Finite Group Theory: Suppose every two generator subgroup of a finite group has a normal $p$-complement. Show that $G$ has a normal $p$-complement. Of course ...
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38 views

Proof that every group of order $15,400$ has a normal subgroup of order $275$

I constructed the groups of order $15,400=2^3\cdot 5^2\cdot 7\cdot 11$ with GAP and noticed that every such group has a normal subgroup of order $275=5^2\cdot 11$. Can this be proven by hand ? ...
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1answer
20 views

Can I get help understanding representations and subrepresentations?

This is in light of the problem posted here. I think I understand the overall idea; we want to essentially equip special vector spaces with groups to gain more insight on the group and what it can do....
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1answer
45 views

Find conjugacy classes of $G= \left\langle a, b \mid a^4, b^2=a^4, aba=b \right\rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
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2answers
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How do I prove symmetry of a relation given a function?

Let G be a group. For all $g\in G$ , define the function f: G → G that sends x to $gxg^{-1}$. Define the relation ~ on G by a~b if $a = f(b)$ for some $g\in G$. Prove that ~ is an equivalence relation....
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1answer
42 views

Is this intuition for the semidirect product of groups correct?

My abstract algebra class introduced me to direct products, not semidirect products. I became interested in semidirect products when confronted with the following homework problem: Define the ...
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1answer
93 views

How many groups of order $2016$ exists, which are a direct product of smaller groups?

There are $6538$ groups of order $2016$ upto isomorphism. How many groups of order $2016$ are a direct product of (at least two) smaller groups ? I calculated an upper bound by summing the ...
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1answer
50 views

$\sum_{d|n} \varphi(d)=n$

I want to solve $\sum_{d|n} \varphi(d)=n$ using Group theory. Here, $\varphi(d)$ is Euler's totient function. I think I should use $\Bbb Z_n$ and fundamental theorem of cyclic group. Then I use $\...
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1answer
28 views

Tate's theorem for Hall subgroups

Let $G$ be a finite group and $P$ a Sylow $p-$subgroup of $G$. Take $P\le V\le G$. Tate's theorem states that if $V\cap A^p(G)=A^p(V)$, then $V\cap O^p(G)=O^p(V)$. Now let $H$ be a Hall $\omega-$...
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52 views

Biggest noncommutative group $N$ where a group $G$ is normal.

Given a finite group $G$, it is possible to obtain the biggest non commutative group $N$ such that $G\lhd N$ with $N\neq G$, $\vert N\vert < \infty$ and $N$ not a direct or semidirect product?
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1answer
46 views

If there is a simple group of order less than 36, then it must have prime order

I want to show that if there is a simple group of order less than 36, then it must have prime order. Is there a quick way to show this or do I have to go through each order 1 though 36 showing that ...
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1answer
45 views

How could compute the centralizers by GAP?

Let $G$ be a finite group. How could we obtain all conjugacy classes of element centralizers of $G$ by GAP? (By the centralizer of an element $g$ in $G$, I mean the subgroup $C_G(g):=\{x\in G | xg=...
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21 views

Construction of a group where words of small length are non neutral

Let $n$ and $p$ be positive integers. Is there a finite group $G_p$ generated by elements $a_1, \dots, a_n$ such that any reduced word on $a_1, \dots, a_n, a_1^{-1}, \dots, a_n^{-1}$ of size $\leq p$ ...