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Questions tagged [finite-groups]

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.

The questions that you can ask under this tag are of the same type that the questions related to the tag : homomorphisms of finite groups, representation theory, structure of finite groups...

You may refer to this Wikipedia entry for an introduction to this topic.

11774 questions
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Examples of groups that are not rings

I am teaching some really advanced high school students about groups and rings and wondering of examples of groups that are not rings. I am hoping to find such examples where addition and multiplication are actually defined for the group. I assume,…
Andrew
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Let $R$ be any fixed rotation and $F$ be any fixed reflection in a dihedral group. Prove that $(FR)(FR)=e$

Let $R$ be any fixed rotation and $F$ be any fixed reflection in a dihedral group. Prove that $(FR)(FR)=e$ I saw this assumption in the back of my textbook for a solution and I don't know how to prove it. It's true of all the little drawings of…
Zaya
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whether a given condition in a theorem is satisfied in general

I have a theorem stating that "a semidirect product of a cyclic group of prime order by an abelian group satisfies a certain property". (The property is given in the theorem and according to the terminology of the text where the theorem is stated it…
Buddhini Angelika
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Why is 9 a generator of a group Z28?

It says in my book that generators are relatively prime to $28$, so that would be a set of $\{1,3,5,9,11,13,15,17,19,23,25,27\}$, ok i get that. But why is $9$ and in there? I can express $9$ as $3^2$, so why we have to put $9$ in there as all the…
Nejc
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When $\pi$-closure equals the normal closure of a subgroup

Let $G$ be a finite group and $\pi$ be a set of primes. For any subgroup $H$ of $G$ define the $\pi$-closure of $H$ as $H^{\pi}=\langle x^{-1}Hx| x\: \text{is a ${\pi}$-element of}\: G \rangle$. Can you help me to find a minimal $\pi$ such that the…
hesim
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Two groups acting iteratively on one space

I have two groups $(Z_2)^n$ equiped with a pointwise binary addition operator and $S_n$ acting iteratively on the space $X=\{0,1\}^n$, in the following way: Fix $x\in X$. At each iteration we apply a transposition from $S_n$ to $x$ and then an…
RezaR
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$g^x = g^{x\bmod m}$ in a finite group of order $m$

I want to understand the proof that $g^x = g^{x\bmod m}$ for $g\in G$ for $G$ such that $\vert G\vert = m$ The proof goes as follows say that $x = qm+r$ then $$g^x = g^{qm+r} = g^{qm}g^r = (g^m)^qg^r = 1^qg^r = g^r = g^{x\bmod m}$$ How is $g^m = 1$?…
user405156
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On the structure of outer automorphism group of $A_6$.

One knows $|Out(A_6)|=4$. Then $Out(A_6)$ is abelian. Furthermore it is cyclic group or element abelian 2-group. QUESTION: IS $Out(A_6)$ cyclic group? Thanks!
C. Simon
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On the Sylow $2$-subgroup of a nonabelian simple group.

Let $G$ be a finite nonabelian simple group and $P$ be a Sylow $2$-subgroup of $G$. QUESTION: Is $P$ cyclic or quaternion group?
C. Simon
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Commutator of groups of order $pq^2$.

Let $G$ be a non-abelian group of order $pq^2$. Is it true that $G'\simeq \mathbb Z_q\times \mathbb Z_q$? From a previous question, I learned that the only non-abelian group of order $pq^2$ is of the type $(\mathbb Z_q\times \mathbb Z_q)\rtimes…
muser17
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Cenraliser, group theory

I would be grateful if someone could comment on my solution to the following question. Let $G$ be a finite group. If $g \in G$ and $g \neq e_{G}$, then prove that $|C_{G}(g)| > 1$. [ $C_{g}$ denotes the centraliser of $g$ in $G$. ] My attempt is…
Gismho
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Minimal normal subgroups of a finite supersolvable groups

Let $G$ be a finite supersolvable group. Then a minimal normal subgroup of $G$ is has prime order. Is it true in general that $G$ has a minimal normal subgroup for each prime divisor of $G$?
R Maharaj
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How to show that $G$ has no element of order $pq$.

Let $G=\langle a,b |a^q=1,b^{p^2}=1, bab^{−1}=a^i, ord_q(i)=p^2\rangle$, here $p^2 \mid q-1$ and $p< q$ are primes and I am assuming that $|G|>12$. Thus $G$ has a normal sylow $q$-subgroup. How can I show $G$ has no element of order $pq$?
user404060
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Characteristic subgroup of a group question

My question: For odd prime $p$, what is the easiest group-theoretic way of showing that $\mathbb{Z}_p^2\rtimes C_2$ has a normal subgroup of size $p$? Attempt: If $\mathbb{Z}_p^2$ has a characteristic subgroup of size $p$, then we are done (since if…
user439872
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order of elements in a finite group

If $|G|=p^rm$ with $(p,m)=1$, suppose that $x\in G$ is an element such that $o(x)=p^{r_1}m_1$ with $r_1>0$ and $(m_1,p)=1$. I dont understand why exist $a,b\in G$ such that: 1) $a$ has order a power of $p$ 2) $b$ has order coprime with $p$ 3)…
Dubious
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