Questions tagged [maximal-subgroup]
To be use for both group theory and semigroup theory.
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questions with no upvoted or accepted answers
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Any non-trivial finitely-generated group admits maximal subgroups
I want to solve the following problem from Dummit & Foote's Abstract Algebra:
This is exercise involving Zorn's Lemma (see Appendix I) to prove that every nontrivial finitely generated group ...
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When does a finite group $G$ only have a single maximal subgroup containing given $A<G$?
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\...
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Is the irreducible $ SU(3) $ subgroup of $ SU(6) $ maximal?
Is the 6 dimensional $ (2,0) $ irrep of $ SU(3) $ maximal in $ SU(6) $?
For those of you who are interested in context, I started wondering this the other day when I tried to write down the maximal ...
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Maximal cyclic subgroups
Can we classify the groups for which every maximal cyclic subgroup is of same order and intersection of any two maximal cyclic subgroups is identity?
For example in case of abelian groups $$G=\mathbb{...
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Why $ K \cap H $ is a maximal subgroup of $H $?
Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal ...
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Groups with all maximal subgroups normal
It is well-known that a finite group is nilpotent if and only if every maximal subgroup is normal. Furthermore every nilpotent group has all maximal subgroups normal.
Question:
Let $G$ be a group ...
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Reduction of structure group to its maximal compact subgroup
I've frequently faced this deduction regarding reduction of structure groups to their maximal compact subgroups. Since each Lie group is product of its maximal compact subgroup and some contractible ...
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Minimal, maximal, least, and greatest element
Let $B = \{1, 2, 3, 6, 12, 18\}$ and $R$ be defined by $xRy$ if and only if $x|y$.
a) Determine all minimal and all maximal elements of the poset.
b) Find all least and greatest elements of the ...
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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 ...
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A question on a 2-group with an elementary Abelian maximal subgroup
Let $G$ be 2-group of order $2^{n+1}$($n\geqslant2$) which has a maximal subgroup $N\cong\mathbb{Z}_{2}^{n}$. It is straightforward to check that if $G$ is an Abelian group, then $G$ is isomorphic to $...
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Every maximal subgroup has prime index.
Let $G$ be finite group, and assume that every maximal subgroup of $G$
has prime index.
Show that $G$ is solvable.
I tried to prove it myself, but I'm afraid there may be mistakes in my proof.
$\...
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If $H$ is a maximal subgroup of $G$, then the action of $G$ on the cosets of $H$ is primitive.
If $H$ is a maximal subgroup of $G$, then the action of $G$ on the cosets of $H$ is primitive.
This is a corollary in my group theory lecture notes with no proof. My thought is the following (I don't ...
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Showing that $\left\langle x - J_\lambda x, x^* - Ax_\lambda \right\rangle \leq \|x^*\|\|x-x_\lambda\| - \lambda^{-1}\|x-x_\lambda\|^2$
Studying Maximal Monotone Operators and the Minty-Browder Theory, I have stumbled accross the following expression which I cannot justify.
It can be found at Page 40, of the book "Nonlinear ...
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Reference for Cardinality of Parabolic Subgroup of Symplectic Group over Finite Fields
I am looking for a source to reference that gives the cardinality of parabolic subgroups of the Symplectic group $Sp$ over a finite field $\mathbb F_q$. What I want is essentially exactly what is in ...
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The Frattini subgroup of the standard Wreath product of two quasicyclic groups is the group itself.
This is part of Exercise 5.2.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE.
Here are some previous questions ...
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Frattini subgroup of $p$-group
Suppose $G$ is a finite, non-trivial $p$-group and $\Phi(G)$ is the Frattini subgroup, defined as the intersection of all maximal subgroups of $G$. Since $G$ is finite, there are finitely many maximal ...
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What are the maximal subgroups of a finite abelian group in terms of characters?
Let $(G,+)$ be a finite abelian group with neutre $0$.
A maximal subgroup of $G$ is a proper subgroup $H$ such that no other proper subgroup $K$ of $G$ contains $H$ strictly.
We know that characters ...
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Intersection of all maximal subgroups of a finite group
Let $G$ be a group. Define
$$F=\left\lbrace x\in G\;:\; \text{if } \left\langle S,x\right\rangle=G\text{ then }\left\langle S\right\rangle=G\right\rbrace.$$
I have that $F$ is a normal subgroup of $G$....
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Why does ergodicity fail?
Apparently the following is not ergodic, and a very sketchy argument is given, so I was hoping someone here would be able to explain/give a better argument.
Suppose $G$ is a compact connected non-...
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2
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Conjugacy classes in maximal subgroups
Let $G$ be a finite group, $H$ a maximal subgroup. If $[G:H] = 2$, it is very well known how to determine the conjugacy classes of elements of $H$: they either stay the same or split depending on ...
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Embedding property of subgroups: Strongly Pronormal
Definition : A subgroup $H$ of $G$ is said to strongly pronormal in $G$, if for each $g\in G$ and for any subgroup $K \leq H$, there exists $x\in \langle H, K^g \rangle$ such that $K^{gx} \leq H$.
It ...
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Maximal (permutation) subgroups of $PSL(2,p)$
I have been trying to look at the maximal subgroups of $\mathrm{PSL}(2,p)$ for $p > 2$ prime and believe I have a sufficiently good idea of how those isomorphic to $C_p \rtimes C_{\frac{1}{2}(p-1)}$...
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Generalization of product theorem in group theory.
I was taught the following group theory theorem:
Let H and K be subgroups of a group G and assume HK = KH
$H \vee K=HK=KH$
which sometimes is called the product theorem (Ledermann, Introduction ...
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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 $CO_{2n+1}$ ...
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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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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 ...
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Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. If $G = Z(G)M$, then $M$ is normal in $G$
I need to prove that if $G$ is a group, $M$ is a maximal subgroup of $G$ and $Z(G) \nsubseteq M$,then $M \unlhd G$. Is true that $G = Z(G)M$, right? Is this enough?
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Frattini subgroup in abelian group
Let $G$ a abelian group. I recall that $G^p=\{g^p:g\in G\}$ ($p$ is primes) and $\Phi(G)$ is Frattini supgroup of $G$ i.e. intersection of maximal subgroup of $G$.
I want proof that $\Phi(G)=\...
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Quotient by maximal normal locally finite subgroup
Any group contains a unique maximal normal locally finite subgroup (this is an exercise in Robinson, second edition, p. 436). Does the quotient by this subgroup have any notable properties?
Edit: ...
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Are there necessary and sufficient conditions for any group to have a maximal normal subgroup?
I know that there are plenty of infinite groups with no maximal subgroups - classic example is the additive group of rational numbers, $\mathbb{Q}$.
Moreover, I know of the result for finite groups ...
user100463
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Groups having no maximal cyclic subgroups of prime order
Is there any classification of groups which do not contain maximal cyclic subgroups of prime order p, for any prime p ?
Your suggestions will be highly useful.
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about almost maximal subgroup
Let $ G $ be a finite group that there is an element $ y\in G $ such that $ G = \langle y \rangle M $for any almost maximal subgroup $ M $ of $ G $ and $ y $ is $ p $-element. ( A proper subgroup $ M $...
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Generalization of a property of finite nilpotent groups
Let $G$ be a finite nilpotent group and $M$ be a maximal subgroup of $G$.
If $H$ is a proper non-trivial subgroup of $G$ such that $H\not\leq M$,
then we can show that $H\cap M$ is a maximal subgroup ...
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