# Questions tagged [maximal-subgroup]

To be use for both group theory and semigroup theory.

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### Counting maximal subgroups of $\mathbb{Z}_m^n$

Let  $$\mathbb{Z}_m=\mathbb{Z} / m\mathbb{Z}$$ How many maximal subgroups does  $$\mathbb{Z}_m^n=\underbrace{\mathbb{Z}_m \times \mathbb{Z}_m \times \cdots \times \mathbb{Z}_m}_n$$  have? (m need ...
• 163
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### Maximal subgroups of $SL(n, 2)$

According to Aschbacher's theorem, if $H$ is a maximal subgroup of $SL(n,q)$, then either it belongs to one of the classes $C_1 - C_8$, or $H$ is absolutely irreducible and $H/(H\cap Z(SL(n,q)))$ is ...
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• 103
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### maximal subgroup of the general linear group

Maybe I'm being silly. As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, is the conformal orthogonal group $CO_{2n}$ a maximal subgroup of $GL_{2n}$(or $CO_{2n+1}$ ...
• 1,283
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### The maximal free abelian subgroup that can be embedded in $GL(n,\mathbb{Z})$

I am stuck on this problem and cannot seem to find a good reason for drawing the required conclusion. The problem is as follows: Given $SL(n, \mathbb{Z})$ a subroup in $GL(n, \mathbb{Z})$. How can ...
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### Primitive maximal subgroups of $S_{n}$

I notice the following result in GTM163. Consider $S_{n}$ act on $\lbrace 1,2,\cdots,n \rbrace$ in a natural way. Then the maximal subgroups $M$ of $S_{n}$ fall into three classes: $(i)$ (intransitive)...
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### G2 as a subgroup of SO(7) from extended Dynkin diagrams?

As it is shown in the post Subgroups of $E_8$ by using extended Dynkin diagrams it is possible to find subgroups of a group from the extended Dynkin diagram by simply cutting nodes. However, the ...
• 218
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### Is there an explicit maximal simple group?

It is not hard to prove the following lemma: LEMMA: Let $(G_i)_{i \in I}$ be a chain of simple groups. Then $G = \bigcup_{i \in I} G_i$ is a simple group. Let $N$ be a normal subgroup of $G$ and ...
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### Let $H\le G,G$ finite. If $M_H$ is a maximal subgroup of $H$, is it the case that $M_H = H \cap M$ for some $M$, a maximal subgroup of $G$?

I have a question regarding finite groups and their subgroups. Specifically, let $G$ be a finite group, and let $H$ be a subgroup of $G$. I am interested in understanding whether the following ...
• 75
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### Maximal subgroup of a finite group contains all other subgroups.

A maximal subgroup $M$ of a finite group $G$ is a proper subgroup of $G$ such that no proper subgroup $H$ exists such that $H$ contains $M$. I want to prove that all other proper subgroups of $G$ ...
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### Do all unitary simple groups $U_{2n+1}(2)$ have maximal subgroups of the form $3^{2n}:S_{2n+1}$?

In the ATLAS, the unitary simple groups $U_5(2)$ and $U_7(2)$ have maximal subgroups of structures $3^4:S_5$ and $3^6:S_7$, respectively. It seems that they are subgroups of the generalized symmetric ...
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### Size of maximal subgroups in the direct product of finite groups

Let $\pi(G)$ be the set of all prime divisors of the order of a finite group $G$. Prove that: if $M$ is a maximal subgroup in $D=G \times G$ then $\pi(M)=\pi(G)$. My attempt: 1) If $G$ is $p$-group ...
• 75
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### $G$ a group with center $\{e\}$ and $A$ a maximal subgroup of $G$ that is also abelian and not normal. How to show that $A$ is a Frobenius complement?

I have been sitting on this homework problem for days now: As the title says, I have a group (which doesn't have to be finite. Even Frobenius groups aren't defined as finite in our course) which only ...
• 33
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### All maximal subgroups are Sylow subgroups

Let $G$ be a group in which all maximal non-trivial subgroups are Sylow subgroups. Then $G$ isn't a simple non-abelian group. I know how to prove this by relying on the theorem that if all maximal ...
• 543
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### is there are a method to build subgroups of the multiplicative group $\left( \mathbb{Z}/n\mathbb{Z}\right)^*$

The multiplicative group $\left( \mathbb{Z}/n\mathbb{Z}\right)^*$ is defined by : $\left( \mathbb{Z}/n\mathbb{Z}\right)^*= \{\, \bar{x} \in \mathbb{Z}/n\mathbb{Z}\;\;:\;\;gcd(x,n)=1 \,\}$ then we ...
138 views

### Algorithm that list all Maximal subgroups (up to conjugacy) of a finite group

Question: Is a polynomial time algorithm known to find all maximal subgroups (up to conjugacy) of a given group? (by polynomial time, I mean polynomial in the input size). If there is, please share a ...
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### Why can’t Zorn’s lemma be used to show any module has maximal submodule [duplicate]

So you can use Zorn’s lemma to show that if $R$ is a ring then $R$ has a maximal left ideal. Show you would let $X$ be the set of all proper left ideals of $R$. Then can show any chain has an upper ...
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### Aschbacher Class $2$ subgroup structure

In $PGL(12,3)$, there should be an Aschbacher Class $2$ subgroup the image of $GL(2,3)^6 \wr{\rm Sym}(6)$. I am trying to locate the image of $GL(2,3)^6$ in Magma using derived subgroup but it doesn't ...
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1 vote
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### Example of a group such that the perfect maximal subgroup is $1$.

I an looking for an example of a group say $G$ such that the perfect maximal subgroup is $1$. I cannot find any references or come up with an example of my own. Could someone please cite a reference?
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### construct the normaliser of a subgroup and then construct the subgroup

There is a maximal nontoral (not contained in a conjugate of a fixed maximal torus) elementary abelian $2$-subgroup of rank 6 in $G = PGL(8,7)$. I denote this group by $A$. Its normaliser $N_{G}(A)$ ...
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### inclusion among finite symplectic groups

This might be a silly question, but do we have the following? $Sp(2,q) < Sp(4,q) < Sp(6,q) < Sp(8,q) <....$ I checked the list of maximal subgroups of $Sp(n,q)$ for $n = 4,6,8,10,12$ and ...
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• 543