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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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Two questions in Isaacs' book Finite Group Theory

I am reading Isaacs' book finte group theory, and I have two questions. in page 90, there is a Wielandt's theorem (if $G$ has a nilpotent Hall $\pi$-subgroup, then all Hall $\pi$-subgroups of $G$ are conjugate), now I know how to prove it, I want…
user94782
  • 241
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Cardinality of $HgK$ with $H,K$ subgroup of a finite group $G$ with fundamental counting principle

Let $G$ be a finite group. Let $g \in G$ and let $H,K$ be subgroup of $G$ I have to compute that $|HgK|=\frac{|H|\cdot |K|}{|H^g \cap K|}$ with the fundamental counting principle. I know that there are some other proof, but I need a proof that use…
Mario
  • 717
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A question on a special automorphism of a finite group

Let $G$ be a finite group and let $\sigma$ be an automorphism of $G$. Generally, if $H$ is a subgroup of $G$, then $\sigma|_{H}$ ($\sigma$ restricted to $H$), is not always an automorphism of $H$. My questions are: If $\sigma\neq id$ and has the…
boaz
  • 4,783
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A question of finite Group

Let $G = A_5$, the alternating group of degree five,. Let $\pi = \{2,3\}.$ Prove that $M$ is a maximal $\pi$-subgroup of $G$ if, and only if, $M\cong A_4$ or $M\cong S_3$, where $A_4$ is the alternating group of degree four and $S_3$ is the…
user59969
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Prove that if $G$ is a finite nilpotent group and $N\trianglelefteq G$, then $N\cap Z(G)\ne\{e\}$.

Prove that if $G$ is a finite nilpotent group and $N\trianglelefteq G$, then $N\cap Z(G)\ne\{e\}$. This is true for all $p$-groups. Suppose $|G|=p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ where $p_1,\cdots,p_s$ are distinct primes, then $G\cong…
Knt
  • 1,649
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Free products of two or copies of a group

Could someone describes the elements of the free product of two (or three...) copies of the finite cyclic group $\mathbb{Z}/N\mathbb{Z}$. Are they words over $\mathbb{Z}/N\mathbb{Z}$ ? Do you allow $0$ (I mean the neutral element of…
fft
  • 21
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Is this sufficient criterion for existence of non-abelian groups also necessary?

In the formula section of Sloane's integer sequence A060652 - Orders of non-Abelian groups , we find Let the prime factorization of $n$ be $p_1^{e_1}\cdots p_r^{e_r}$. Then $n$ is in this sequence if $e_i>2$ for some $i$ or $p_i^k \equiv 1 \pmod…
Hagen von Eitzen
  • 374,180
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Lemma for 1st Sylow Theorem

If $p^f$ is the largest power of a prime $p$ such that $p^f|n$, then $$p\text{ does not divide }n\text{ choose }p^{f}\text{ (not sure on syntax for combinatorials).}$$ The proof in Artin's Algebra is outlined by this claim, which I haven't been able…
roo
  • 5,598
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Expression for a subgroup of a inner direct product group.

Suppose I have a group $G$ which is the (inner) direct product of subgroups $H$ and $K$. That is, $G = HK$, $H\cap K = \{1\}$, and $H$ and $K$ are both normal subgroups of $G$. Then for a subgroup $G'$ of $G$, I want to write (in fact a proof I'm…
roo
  • 5,598
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Number of involutions in a finite group

It is easy to see that the number of involutions in an elementary abelian $2$-group of finite rank $n$ is $2^n-1$. Is there a formular for computing the number of involutions in any finite abelian group or precisely, is there any bound on the number…
Chuks
  • 1,241
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Groups of certain order

Let $G$ be a finite group of order $2^aq^b$ for odd prime $q$ and $a,b\geq 2$. Suppose $G$ has at least $2$ subgroups $A$ and $B$ of order $2^{a}$. I suspect that $A\cap B$ is a normal subgroup of size $2^{a-1}$. Any hint to the proof or give a…
INT
  • 21
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Subgroups of Index 2

True/False: Every finite abelian group of even order has a subgroup of index 2. There exists an element of order 2, and hence a subgroup order 2 (Call it $H$). Let H = Ker$\phi$ where $\phi:$ $G \rightarrow G/H$. I was trying to use the fact that…
sobrio35
  • 111
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For which numbers there is only one simple group of that order?

There is only one simple group of orders: 3, 60, and 360 respectivley. Are there other groups of this kind? What general characteritics do they share? From pure curiosity did this question arise. Thanks for any help.
awllower
  • 16,536
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Find an inverse element for an element in this Group

We know that if $\Omega$ be set of all 1-dimension subspaces of $V=V_{2}(q)$ which $V$ is a vector space on a finite field $GF(q)$ and so $|V|=q^{2}$ then, group $PGL_2(q)$ acts on $\Omega$. Also, it can be proved that the below set $G$ is a group…
Mikasa
  • 67,374
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Is this true that $p^2\big||Aut(G)|$?

Is this statement true that for a finite and non abelian $p$-group $G$; $p^2\big||Aut(G)|$? I just found $Q_8$ fulfillment of this claim.
Nancy Rutkowskie
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