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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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 ...
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1answer
50 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 ...
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2answers
41 views

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 ...
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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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25 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
21 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 ...
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0answers
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 ...
2
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1answer
79 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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0answers
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
63 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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0answers
42 views

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 ...
3
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1answer
44 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 ...
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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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0answers
20 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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32 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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0answers
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 ...
4
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1answer
67 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 ? ...
2
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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 ...
3
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1answer
44 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, ...
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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 ...
2
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1answer
41 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
88 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
26 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 ...
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2answers
51 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?
4
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1answer
43 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 ...
2
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1answer
42 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 | ...
2
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0answers
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 $\leq p$ ...
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2answers
200 views

Show that the group is not simple

I want to show that: If $G$ contains a subgroup with index at most $4$ and $G$ has not a prime order, then $G$ is not a simple group. Let $N\leq G$ with $[G:N]\leq 4$. We have that ...
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0answers
37 views

Product of Cycles: Name to denote “direction” of composition

Is there a notation to denote the difference between these two products of cycles? It seems as though there are two conventions out there that should have a specific name for them. The subscripts for ...
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Cyclic Groups and prime order

The following example is given in my textbook: Example: If $G$ is a group and $|G| = p^2$ then either $G$ is cyclic or $g^p = 1$ for every element $g \in G$ Solution: Assume that $G$ is not cyclic. ...
5
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1answer
73 views

How many groups of order $512$ and $1024$ are there with a center of size $2$?

I did not find a sequence in OEIS about the number of groups of a given order with a center of size $2$. For the first few powers of $2$, the numbers are : $2$ : $1$ group $4$ : $0$ groups $8$ : ...
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23 views

Generators of the group $2^6 \rtimes 3 \cdot S_6$ in the Miracle Octad Generator (MOG)

I am studying the large Mathieu groups and more specifically I have arrived at the Golay code and the Miracle Octad Generator. My question comes from Robert Wilson's book "The Finite Simple Groups". ...
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2answers
41 views

Any two Singer cyclic subgroups of GL(n,q) are conjugate

Cyclic subgroups of $\operatorname{GL}(n,q)$ of order $q^n - 1$ are called Singer cyclic subgroups. The following statement seems to be well-known: Any two Singer cyclic subgroups of ...
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Possible natural arguments: to show subgroup of order $m$ in group of order $2m$, $m$ odd

A simple interesting result in Group Theory on the existence of (normal) subgroup is following: If $|G|=2m$, $m$ odd, then $G$ has a (normal) subgroup of order $m$. This theorem, as a ...
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1answer
64 views

If $G$ is a group of order $4n+2$, then $G$ contains a subgroup of order $2n+1$ [duplicate]

This question comes from M.A. Armstongs book Groups & Symmetry, Chapter 13, #5. I came across a similar question here, but I am not familiar with the language yet and would rather solve using the ...
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1answer
55 views

Sylow subgroup of some factor group.

Let $G$ be a finite group. Let $K$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. Let $P$ be a Sylow $p$-subgroup of $K$. Is $PN/N$ is a Sylow $p$-subgroup of $KN/N$? Here is what I ...
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0answers
25 views

Calculating amount of $2$-sylow subgroups of $S_{2^n}$.

Main question: How do I calculate the number of $2$-sylow subgroups of $S_{2^n}$? Let $n \in \mathbb{Z}_{\geq 2}$. I have a $2$-sylow subgroup $H \subset S_{2^n}$ (too long to spell out all the ...
3
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1answer
70 views

Given the generators, find the entire group

my question is quite simple. If you are given the generators of a group, is there any systematic way to generate all of the elements of the group? For example, suppose that you have the Hadamard ...
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1answer
39 views

The “obvious” symmetry group $C_3 \times S_4$ related to the hexacode

I am studying the large Mathieu groups and more specifically the hexacode from Robert Wilson's book "The Finite Simple Groups". The following paragraph is from page 184:The hexacode My question is ...
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2answers
78 views

$|G|=|H_1| |H_2 |$ and $H_1 \cap H_2 = e $. Is $H_1 H_2 =G$? [closed]

If there is a group $G$ with order $a$, having a subgroup $H_1$ with order $b$, and $H_2$ with order $c$, and $bc=a$, $H_1 \cap H_2 = e $. Is $H_1 H_2 =G$?
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0answers
25 views

Right coset of a subgroup H in G

Let $H = \left \{(1),(12)(34),(13)(24),(14)(23)\right \}$ be a subset of $G = S_4$. Observe that $H$ is a subgroup of $G$. Determine the set of right cosets of $H$ in $G$. Let $K=G_1$ and show that ...
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1answer
43 views

Find a group of order 60 with subgroup of all possible orders

I am asked the find the possible orders of subgroups of a group of order 60. By Lagrange's theorem: $\left | H \right | | \left | G \right |$ Any positive integer n that is a divisor of $\left | G ...
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1answer
28 views

All generator of a subgroup of finite order n

Let $G = \left \langle g \right \rangle$ be a cyclic group of order 20 Find all generators of the subgroup of order 10? At this point, I refer to the corollary: Let $G = \left \langle g ...
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1answer
36 views

$G \cong \mathbb{Z}_{p^{n_1}}\times \dots \mathbb{Z}_{p^{n_k}}$. If $p=2$ and $n_1>n_2$, prove that $L(G)\cong \mathbb{Z}_2$.

Let $G \cong \mathbb{Z}_{p^{n_1}}\times \dots \mathbb{Z}_{p^{n_k}}$ be a finite abelian $p$-group, in which $n_1\geq \dots \geq n_k$. Define $$L(G)=\{g\in G \;|\;\alpha(g)=g\; ,\forall \alpha\in ...
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1answer
47 views

Counter-intuitive practice problem to group and operation

Determine if $\ast$ is a binary operation and if$ \left ( G,\ast \right ) $is a group. Explain. Question: $G=\mathbb{N}$ and $a\ast b$ is the smallest integer greater than both a and b. For ...
2
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0answers
35 views

$G=H\times K$ and $H,K$ have no common direct factor but $H$ or $K$ is not characteristic in $G$

I was searching for an example of a group $G=H\times K$ where $H,K$ have no common direct factor (i.e. there is no $L\neq 1$ such that $H\cong H_1\times L$ and $K\cong K_1\times L$), but $H$ or $K$ is ...