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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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Where to look up numerical information of finite groups of small orders (<2047)?

I need some information of finite groups of order (<2047) for my thesis project, such as the number of their normal subgroups, the composition length and composition series. And I also need to carry out some numerical experiments, for example…
youknowwho
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finite groups $G$ such that any two commuting elements of $G$ lie in some cyclic subgroup of $G$

If $n$ is odd, the symmmetry group of a regular $n$ gon (also called a dihedral group) has the property that given two commuting elements $a,b$ it follows that $a,b$ lie in a cyclic subgroup. [This includes an element commuting with its inverse,…
coffeemath
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A finite simple group with $\pi (G)\subseteq \pi (p^{2}-1)$

Let $\pi (k)$ be the set of prime divisors of $k$ and let $\pi (G)=\pi (|G|)$. Let $G$ be finite simple group with $\pi (G)\subseteq \pi (p^{2}-1)$, where $p$ is prime. I would like to know is there any classifications for group $G$?
user2132
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Suppose $G$ is a group of order $n$ . Let $p$ and $q$ be distinct primes which divide $n$. Can we say that $G$ has a subgroup of order $p\cdot q$?

Suppose $G$ is a group of order $n$ . Let $p$ and $q$ be distinct primes both of which divide $n$. Can we say that $G$ has a subgroup of order $p\cdot q$?
user566574
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Is it true all centralizer of G are abelian?

Suppose $G$ is a finite group such that $\frac{G}{Z(G)}\cong Z_p\times Z_p\times Z_p$. Is it true all centralizers of G are abelian?
m a s
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An Exercise of Finite Groups

Prove that the group $F(2,5)$ given by $$\left$$ is a finite cyclic group.
user59969
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Schur multiplier of direct product

Let $A$ and $B$ be arbitrary finite groups with $(|A|,|B|)=1$. Let $M(G)$ be the Schur multiplier of the group $G$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $M(A \times B) \cong M(A) \times M(B) $. We have that $|M(A…
cas
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Exponent of a finite group $G$

Let $G$ be a finite group. The exponent of $G$, $\exp(G)$ is defined as the minimal positive integer $m$ such that $x^m=1$ for all $x \in G$. prove: $a)$ if $G$ is abelian then $\exp(G)= \max\{\text{ord}(x):x \in G\}$ $b)$ if $G$ is not abelian the…
user614287
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Mean in multiplicative group modulo n

How can I calculate the arithmetic mean in a multiplicative group modulo $n$? For example: if $n=15$, $a=8$ and $b=5$ then I want to find the mean of $a$ and $b$ as $7$. If $a$ and $b$ were real numbers then, the mean of $a$ and $b$ can be defined…
mip
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Number of ways to color cube sides.

How many ways can cube sides be colored by 4 types of color? I need something, not exactly the answer, maybe just formulas or theory I need. Up to 4 different colors. no, rotation does not change.
Vlad Trotsenko
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Equivalent definition of the ord$(g)$

Define the order ord$(g)$ of an element $g$ in a finite group $G$ to be $|\{g^0,g^1,g^2,\ldots\}|$. I want to prove an alternate definition of the order of an element, but I don't know how to prove this implication: Show for an arbitrary $g\in G$…
Guest
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group: $\forall a\in G, \exists n\in\mathbb{N}, a^n=e$

Let $(G,\cdot, e)$ a finite group. Prove that for all $a\in G$, exists $n\in\mathbb{N}$ such as $a^n=e$. Intuitively I understand that there should be equivalence departments according to the generator organ but I don't know to translate it to math,…
J. Doe
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Prime divisor in a finite group

If a prime number $p$ divide the number of elements of order $k$ (for some $k\neq p$) in a finite group $G$, then whether we can say that $p$ divide order of $G$?
Simon
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Element of certain order in special linear space.

What would be the conditions (if any) on the trace of an element in $SL(2,p)$ in order for it to have order 5 ? (assuming $p= \pm1 mod 10$) For example, any traceless element in $SL(2,p)$ has order 4 (straightforward proof). Any suggestion or…
S_j
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Using normal series of a group to build it up via extensions

In the context of normal and subnormal series I've found the following: "From a finite subnormal series of a group $G$ we obtain a sequence of exact sequences and thus $G$ is built up out of the quotients factors of the sequence by forming…
Matt S.
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