Questions tagged [maximal-subgroup]

To be use for both group theory and semigroup theory.

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On the number of invariant Sylow subgroups under coprime action -Antonio Beltrán, Changguo Shao

I'm reading the papers of Antonio Beltrán, Changguo Shao. The article is On the number of invariant Sylow subgroups under coprime action: https://www.researchgate.net/publication/...
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example of a non-abelian group $G$ and a non-trivial maximal normal subgroup $N$ so that $[G : N] ≥ 3$.

I am looking at non-abelian and non-trivial maximal normal subgroups whose indexes are greater than or equal to $3$. I can't find any examples of this anywhere. Could someone give me an example of ...
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Let $p$ be a prime number and let $G$ be a finite $p\text{-group}$. Let $M$ be a maximal subgroup of $G$.

QUESTION: Let $p$ be a prime number and let $G$ be a finite $p\text{-group}$. Let $M$ be a maximal subgroup of $G$. Show that $M$ is a normal subgroup of $G$ and that $| G: M | = p$. THE HINT GIVEN ...
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Let X := {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43} be ordered by divisibility. Find the maximal and minimal elements of X. [closed]

What does it mean that X is ordered by divisibility when all elements of X are prime numbers? Also doesn't there have to be some partial ordering relation for there to be maximal and minimal elements? ...
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Number of maximal subgroups in finitely generated amenable groups

The following statement is known to be true: Any subgroup of a finitely generated group lies in a maximal subgroup Proof: Suppose, $G = \langle \{x_1, … , x_n\} \rangle$ is a counterexample. Then ...
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Uniqueness of maximal subgroup and order being a power of a prime

Let $G$ be a finite group. If $G$ has only one maximal subgroup (a maximal subgroup is a proper subgroup $M$ that given a subgroup $H$ of $G$, $M \subset H \subset G$ implies that $H = M$ or $H = G$), ...
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The Frattini subgroup of $\Bbb{Z}_p \times\Bbb Z _{p^2}.$

Can anyone please help me to find the Frattini subgroup of $\mathbb{Z}_p \times \Bbb Z _{p^2}$? I know that as a set the Frattini subgroup is the set of all non-generators. Is this the only way to ...
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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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Maximal subgroup of $S_n$

Let $S_n$ denote the symmetric group on $\{1,\ldots,n\}$. Let $M$ be the subgroup $\{\sigma \in S_n \mid \sigma(1) = 1\}$. Show that $M$ is a maximal subgroup of $S_n$. Here is what I've come up ...
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Normalizer of a solvable maximal subgroup

Let $G$ be a finite group and let $H$ be a solvable maximal subgroup of $G$, meaning that the only solvable subgroup of $G$ containing $H$ is $H$. I am trying to prove that $H=N_G(H)$. Since $H$ is ...
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Under what assumptions will $M \unlhd G$? ($M$ maximal)

Let $G$ be a finite group (non-trivial), and let $M$ be a maximal subgroup of $G$. My question is, what criterion on $M$ would allow us to deduce that $M \unlhd G $? (I mean, will $M \unlhd G$ ...
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Characterising maximal subgroups of cyclic groups.

Suppose $G = \langle x \rangle $ is a cyclic group of order $n\geq 1$. Prove that a subgroup $H \leq G$ is maximal if and only if $H = \langle x^p \rangle$ for some prime $p$ dividing $n$. Source ...
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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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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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Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...
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Understanding proof that exponential map of compact connected Lie group is surjective

Let $G$ a compact connected Lie group. Then, the exponential map $\exp: LG \rightarrow G$ is surjective. (where $LG$ is the Lie Algebra of $G$). $\textbf{Proof:}$ For any torus $T' \subset G$ we have ...
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Let $G$ be a non-abelian finite $p$-group, do we have $Z(G) \leq \Phi(G)$ in general?

Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) \leq \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I ...
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345 views

Let $G$ be transitive on $S$. Show that the action is primitive if and only if every $\operatorname{Stab}_G(a), a\in S$, is a maximal subgroup of $G$.

I am self-studying "Classical Groups and Geometric Algebra" by Larry C. Grove. This is the 2nd question of the exercises of the 0th Chapter. Let $G$ be transitive on $S$. Show that the action is ...
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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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Why $a,b$ assumed coprime?

I have a question about an exercise in Dummit-Foote Abstract algebra: This is the exercise-solution I am referring to: My question is about problem 3's solution of $\implies$ direction. I am ...
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How can we describe all maximal proper subgroups of $G \times G$

Suppose G is a finite group, $\mathfrak{M}_G$ is the set of all its maximal proper subgroups. Is there any way to describe $\mathfrak{M}_{G \times G}$ - the set of all maximal proper subgroups of $G \...
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Prove $\hat{G}\rtimes Aut(G)$ is primitive when $G$ is characteristically simple

Define Hol(G) as $\hat{G}\rtimes Aut(G)$, where $\hat{G}$ means the group of right multiplication induced by $G$. The problem is to prove Hol($G$) is primitive when $G$ is characteristically simple. ...
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Example of a group where a proper subgroup is not included in a maximal element

So I am trying to figure out if this is possible or not: Given $G$ a group where there exists a maximal subgroup $M$, is it possible for $G$ to have a subgroup $H$ where $H$ is not contained in any ...
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$(\mathbb{Q},+)$ has no maximal subgroup.

Definition. A maximal subgroup of a group G is a proper subgroup $M$ of $G$ such that there are no subgroup $H$ of $G$ such that $M<H<G$. Now, I want to solve the following problem: Problem: ...
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$G$ is a group of order $pq$ and $P_q$ and $P_p$ are Sylow subgroups…

If $(G,*)$ is a group of order $pq$, then it is clear that there are Sylow subgroups $P_q$ and $P_p$ of order $q$ and $p$ in $G$. If $q>p$ then $P_q$ is normal. I want to find a decomposition for ...
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Generating Prüfer 2-group without its Frattini Subgroup

I must be reading this wrong; please help me! It seems to say we can always remove the non-generating elements from any of a group's Generating Sets and still have a Generating Set. But it also ...
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Number of nonidentity Elements Contained in Conjugates of $M$

Prove that if $M$ is a maximal group of $G$ that is not normal in $G$, then the number of nonidentity elements of $G$ that are contained in conjugates of $M$ is at most $(|M|-1) |G:M|$. Here is my ...
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Find the maximal subgroups of $Z/nZ$

The problem is to find the maximal subgroups of $Z$ and $Z/nZ$. I did the first part and I found the maximal subgroups of $Z$ are: $nZ$ with n as a prime integer. I'm a little bit confused with ...
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Subgroup maximal in its normalizer

Let $G$ be a finite group, $H$ a subgroup of $G$, such that $H$ is a proper subgroup of $\operatorname{N}_G(H)$. Is there a sufficient condition for $H$ to be maximal in $\operatorname{N}_G(H)$? What ...
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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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Index of a maximal subgroup in a solvable group

Let $p$ be a prime and $n$ be a positive integer. Is it possible to find a finite solvable group $G$ with a maximal subgroup $M$ such that $|G:M|=p^n$? If $n=1$, we can surely find it taking a group ...
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Intersection between two conjugates of a maximal subgroup.

Let $G$ be a non-abelian finite group such that every proper subgroup of $G$ is abelian. Suppose $M$ is a maximal subgroup of $G$ which is not normal in $G$. I was asked to show that $\bigcup_\limits{...
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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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Does a maximal subgroup have to contain the center?

Let $G$ be a non-abelian group with center $Z(G)$, and let $H$ be a maximal subgroup of $G$ which is non-abelian. Is it true that $Z(G)\leq H$? I am trying to prove that it is true by arguing that if ...
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Why $\mathbb{T}^1$ is maximal in $SU(2)$

This is a question from Stillwell's "Naive Lie Theory" (3.5.2) which I am self studying. There is a less explicit form of the question was asked here, but I don't fully grasp the answer and would ...
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$|G|=p^n$, $Z(G)=\langle w\rangle$, $Z(G)\subseteq$ a maximal subgroup; then $\exists f\in \operatorname{Aut}(G)\ , k\ne1\pmod p$ s.t. $ f(w)=w^k$?

Let $G$ be a finite $p$-group , let $Z(G)=\langle w\rangle$, suppose there exist a maximal subgroup of $G$ containing $Z(G)$; then does there exist $f\in \operatorname{Aut}(G)$ such that $f(w)=w^k$ ...
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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 ...
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Maximal & Maximal Normal Subgroups

I have to answer the following two questions: Using Zorn's Lemma, one could try to give a "proof" of the following statement: Every subgroup $H$ of a group $G$ such that $H \neq G$ is contained in a ...
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GAP/Magma-cas: Suppose $H<S_n$ (given by generators): Does either system make it easy to find the maximal subgroup containing $H$?

I am not sure that this is the right forum, but anyhow: Suppose I have a subgroup $H$ of $S_n$ (given by generators). Does either system make it easy to find the maximal subgroup containing $H$?
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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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Stabilisers in non-abelian $p$-groups are contained in non-transitive maximal abelian subgroups?

Let $P$ be a non-abelian finite $p$-group which is a transitive permutation group of degree $p^n$ such that the stabiliser of a point is meet-irreducible. Suppose $P$ has a non-transitive abelian ...
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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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Intersection of Maximal subgroups

Let G be a group. Show that the intersection of all maximal subgroups of G is a normal subgroup. I proved that the normalizer of a Maximal subgroup is either the subgroup itself of the Maximal ...
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If we have that H<G and |H|=|G| does this imply that H=G?

I have a question i am trying to prove that if $H<G$ and $\dfrac{|G|}{|H|}$ is a prime number then H is a maximal subgroup. I prove this by contradiction, thus i assume that $\exists K : H<K&...
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Why is $A_n$ maximal in $S_n$?

I am struggling to see why exactly $A_n$ is maximal in $S_n$. A subgroup $M$ of a group $G$ is called a maximal subgroup if $M\neq G$ and the only subgroups of $G$ which contain $M$ are $M$ and $...
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All maximal independent sets of a matroid have the same cardinality

How to prove that all maximal independent sets of a matroid have the same cardinality. Provided a matroid is a 2-tuple (M,J ) where M is a finite set and J is a family of some of the subsets of M ...
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cardinality of a maximal subgroup of a $p$ group

Let $P$ be a $p$ group with $|P|=p^n$. Let $M$ be a maximal subgroup of $P$. Is it true that $|M|=p^{n-1 }$ ?
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how to check if a subgroup is maximal?

Is there any general strategy to check whether a subgroup is maximal or not ? For example, in case of rings, we know that an ideal $I$ of a ring $R$ is maximal if and only if $R/I$ is a field. Is ...
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When is the commutator subgroup a maximal subgroup? [closed]

Let $G$ be a group , under what conditions do we have that $G/[G,G]$ is a finite group of prime order ?