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Questions tagged [maximal-subgroup]

To be use for both group theory and semigroup theory.

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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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1answer
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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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Display the values of Christoffel symbols in simplified form in Maxima software

I have calculated all Christoffels symbols (my metric is symmetric type) by hand and now I am trying a lot to compute all of the non-zero values of Christoffel symbols by maxima just for confirmation ...
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1answer
139 views

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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1answer
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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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1answer
58 views

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 ...
3
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1answer
41 views

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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1answer
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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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1answer
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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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2answers
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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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1answer
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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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1answer
93 views

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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1answer
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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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1answer
29 views

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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2answers
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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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1answer
126 views

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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1answer
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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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2answers
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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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1answer
163 views

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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1answer
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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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1answer
207 views

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 ...
3
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1answer
198 views

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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58 views

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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1answer
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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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2answers
392 views

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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1answer
81 views

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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4answers
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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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1answer
436 views

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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2answers
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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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1answer
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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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166 views

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 ?
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1answer
147 views

example of a group which has no proper subgroup of finite index, but it does have maximal subgroups.

Let G be a group. if G has no proper subgroup of finite index, can we say that it has no maximal subgroup? if it is not true, what's the counterexample for this assertion?
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2answers
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In a finite p-group,H is a maximal sub group iff H is normal in G and |G:H|=p

Let G be a finite p-group,H is a maximal subgroup of G if and only if H is normal in G and |G:H|=p I tried acting H on right cosets of H in G .... I don't know what to do now...
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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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1answer
497 views

$G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate

Let $G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate and any proper subgroup is contained in a maximal subgroup . Then is $G$ cyclic ? I ...
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1answer
94 views

Self normalizing maximal subgroup of a non solvable group

Let $G$ be a finite non-solvable group and $H$ its maximal subgroup. Prove that if $H$ is solvable then $H=N_G(H)$ I think I found different ways to prove it but I don't know how to begin: -if $N_G(H)=...
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1answer
80 views

Is the maximal torus a conjugacy class?

Let $G$ be a compact Lie group and consider $T$ a maximal torus in $G$. At Wikipedia I've read that $T$ is a conjugacy class of subgroups of $G$. Does it means that there exist $t \in G$ such that $T ...
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2answers
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If $H \triangleleft G$, and $H$ is maximal, then $G/H$ is simple

I would appreciate it very much if you could take a look at my proof and tell whether or not it's sufficiently good. Proof: Suppose there exists $N \triangleleft G/H$, then $(aH)(nH)(aH)^{-1}=nH=(...
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1answer
29 views

$M$ is normal if $Z\not\subseteq M$

Let $G$ be a group. Let $M$ be a maximal subgroup of $G$. I have to solve the following questions: (a) Prove that $M$ is normal if the center $Z$ of $G$ satisfies $Z\not\subseteq M$. (b) Prove $G/M$ ...
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1answer
318 views

An infinite group with no infinite sub groups

Let $G$ be an abelian group. If all proper subgroups of $G$ are finite, what can we say about $G$. Are the any properties, $G$ is guaranteed to have.
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50 views

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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1answer
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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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1answer
102 views

property about centralizer of maximal subgroup

How we can show that for group $G$ (finite non-abelian p-group, I don't know which ones are necessary) and $M$ maximal subgroup of $G$. We can have $C_G(M)\le C_G(\Phi(G))\le Z(\Phi(G))$ $\Phi(G)$ ...
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1answer
94 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
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1answer
44 views

$ K/N $ be a normal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? [closed]

Let $ G $ be a finite group and $ N $ is a normal subgroup of $ G $. Suppose $ K/N $ be a maximal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $?