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.

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Homomorphism and images

G is a finite group, $\phi:G \to G$ a homomorphism. $\psi:G \to G$ is a homomorphism defined by $\psi(x)=\phi(\phi(x))$. Prove that $(\ker\phi= \ker \psi)\implies($Im$ \psi=$Im$ \phi)$. Can someone ...
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Construction of Subgroups of $S_n$ of a Certain Size

I am interested in constructing a subgroup of $S_n$ of size on the order of $\Theta(\sqrt{n!})$. The algorithm to construct such a subgroup should ideally also take around $O(\sqrt{n!})$ time. One ...
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1answer
51 views

Permutation(?) mapping [duplicate]

Problem Statement: Let $G$ be a finite group, say a group with $n$ elements, and let $S$ be a nonempty subset of $G$. Suppose $e \in S$, and that $S$ is closed with respect to multiplication. ...
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Minimal finite groups with a given simple factor

Let $S$ be a non-abelian finite simple group. Call a finite group $G$ $S$-minimal if it admits $S$ as a Jordan-Hölder factor, but no proper subgroup of $G$ admits $S$ as a Jordan-Hölder factor. For ...
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For $H\lhd G,$ is it true that $O_{\pi}(H)\le O_{\pi}(G)$?

Let $\pi$ be a set of primes, and a $\pi$-group is defined as a finite group with each prime divisor of the order of the group is contained in $\pi$. Let $O_{\pi}(G)$ denotes the unique largest ...
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Define centralizer of chief series

Let $K \unlhd G$ and $ K \leq H \leq G$. The Centralizer of $ H/K$ in $G$ is defined to be the subgroup $J$ of $G$ such that $ K \leq J$ and $ J/K = C_{G/K}(H/K)$. Also we write $J = C_G (H/K)$. Then ...
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The permutizer of a subgroup H of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with H

The permutizer of a subgroup H of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with H, i.e $P_G(H)=\langle x\in G \mid \langle x \rangle H = H \langle x ...
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Notation of Burnside's group theory book “The theory of finite groups”

According to the classification of finite non-abelian groups of order $p^4$ in Burnside's book "The theory of finite groups, 1897, pages 87-88" one of the types of these groups is the following ...
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2answers
72 views

Example of an infinite abelian group having a non-cyclic finite subgroup [closed]

Give example (if exists) of an infinite abelian group having a non-cyclic finite subgroup . Please help
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groups of order $p^4$

I need the classification of finite non-abelian groups of order $p^4$ from E. Schenkman's book "Group theory, 1965". Unfortunately our library has no this book and there does not exist the full ...
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Proving a direct product of groups is a group

I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance. Let $n \geq 1$ be an integer. ...
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Can I recover a group by its homomorphisms?

There is finitely generated group $G$ which I don't know. For every finite group $H$ I can think of, I know the number of homomorphisms $G \to H$ up to conjugation. (By this I mean that two ...
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3answers
39 views

Abelian group that has power of prime order has an element whose order is power of prime

If a finite abelian group has order a power of a prime p, then the order of every element in the group is a power of p. Hi I used Lagrange's theorem that order of element in Group (order of ...
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3answers
51 views

Finite group $G$ has a generating set with following condition.

Let $G$ be a finite group. Prove that $G$ has a generating set $\Omega$, with $|\Omega| \leq \lfloor \log_2 \lvert G \rvert \rfloor$. Thanks in advance.
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the span of a representation's action on a vector

Consider the image of the action of a group representation $\rho: G \to V$ on some vector $v \in V$: $$ \{ \rho(g) v : g \in G \} $$ It seems that the span of this set: $$ W_v \equiv ...
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2answers
71 views

If $m$ is a divisor of $|G|$, then $G$ contains an element of order $m$

If $G$ is a finite group and $m$ is a divisor of $|G|$, then $G$ contains an element of order $m$. I know this is false, but why? Am I supposed to use Lagrange's theorem?
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How i calculate p-Fitting subgroup of $S_3$?

A finite group $G$ is said to be $p$-nilpotent (where $p$ is a prime) if it has a normal Hall $p'$-subgroup, that is, if $O_{p'p}(G) = G$. Obviously every finite nilpotent group is $p$-nilpotent; ...
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about 2-sylow subgroups of group of the form $G \times H$

suppose $G$ and $H$ are finite groups,consider the group $G \times H$ ,and suppose $G_1$ is 2-sylow subgroup of $G$ and $H_1$ is 2-sylow subgroup of $H$ , does $ G_1 \times H_1$ is 2-sylow subgroup ...
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1answer
44 views

Automorphism group of the general affine group of the affine line over a finite field?

I am wondering what the structure of the automorphism group of the general affine group of the affine line over a finite field looks like. I'll make that a bit more precise: If $k$ is a finite field, ...
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What is known about the numbers $M_p = \left\vert C(\mathbb{F}_p )\right\vert$?

There is a question (2.4.c) marked ** (to denote "extremely difficult/currently open problem") in Silverman and Tate's Rational Points on Elliptic Curves which I found really interesting and wondered ...
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Prove that group of symmetries is isomorphic to $S_n$

In my algebra book the first section has the following exercise: Prove that group of all symmetries(isometric bijections under composition) of a regular tetrahedral is isomorphic to $S_4$. I did it ...
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1answer
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Can a normal subgroup of a finite nonabelian group be nonabelian?

We know that if a group is Abelian, then all its subgroups are normal. Also, if a group is nonabelian, it can contain a subgroup which is Abelian. Eg: The Dihedral group of order 2n, $D_{2n}$ is ...
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Converse of Lagrange's Theorem

I want to know partial converse of Lagrange's theorem is true upto how much ? We know that it holds only in case of cyclic groups. Also if a group has order $p^m\times n;\gcd(m,n)=1$ then it has a ...
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How to find the number of non-isomorphic groups of order 10?

How to find the number of non-isomorphic groups of order 10? Using Cauchy I can say that it has an element of order 2 and an element of order 5,and so one group that I can manage is $\mathbb Z_{10}$ ...
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Classifying groups of order $8$

Given group $T$ of order $8$, and $t \in T$ such that $ord(t) = 4$. Let $P = \{1,t,t^2,t^3 \}$ and let $x \in T−P$. List possibilities for $x^2$ labelling as $(a_1,a_2, \ldots ,a_n)$. List ...
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$G$ is a super soluble group and $H$ is a subgroup of $G$ and $\frac{K}{L}$ is chief factor of $G$ . show that $HK=HL$ or $H \cap K=H \cap L$ .

suppose that $G$ is a super soluble group and $H$ is a subgroup of $G$ and $\frac{K}{L}$ is chief factor of $G$ . then show that $HK=HL$ or $H \cap K=H \cap L$ . any hint or Idea will be ...
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1answer
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$|D_{18}|=18=2 \times 3^2$, then $D_{18}$ have 2-sylow and 3-sylow subgroups.

Let $G=D_{18}=\langle a , b | a^9=b^2=1 , bab=a^{-1} \rangle$. Then $D_{18}=S_3 \times Z_3$? $|D_{18}|=18=2 \times 3^2$, then $D_{18}$ have 2-sylow and 3-sylow subgroups. 3-sylow subgroup of ...
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a question about p-super soluble group $G$.

suppose that $G$ is a p-super soluble (p is a prime number) and $H$ is a p-subgroup of $G$ and $\frac{K}{L}$ is a chief factor of $G$ ,then show that $HK=HL$ or $H \cap K=H \cap L$ . definition of ...
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Number of the subgroups of a $p$-group with order $p^k$ is congurent to $1$ modulo $p$

Let $G$ be a $p$ group of order $p^n$ and $k\leq n$. Theorem:Number of the subgroups of $G$ with order $p^k$ is congurent to $1$ modulo $p$. I have found a proof of this theorem in Rotman's book ...
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Show there is no non-abelian group of order 9 [duplicate]

I want to show there is no non-abelian group of order 9. How should I attempt this?
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Non-abelian finite group in which more than half of the elements have order $2$

Is there an non-abelian finite group, in which more than half of the elements have order $2$ I only know that if there is one, then all elements (except identity) cannot have order $2$, otherwise ...
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Fill in a group table with $4$ elements

There is exactly one group $G$ of four elements, say $G = \{e, a, b, c\}$ satisfying the additional property that $xx = e$ for every $x \in G$. Complete the following group table of $G$. $$ ...
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$G/Z(G)$ is cyclic then is abelian?

my question is if $G/Z(G)$ is order of p then is commutative then is abelian group but if G is abelian then $G=Z(G)$ therefore $G/Z(G)$ is not order of p?Is it contradiction?
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Show that if $n$ is not prime then $\mathbb{Z}/n\mathbb{Z}$ is not a field

I read book of Dummit and Foot Abstract algebra. I need some help with the following question. Show that if $n$ is not prime then $\mathbb{Z}/n\mathbb{Z}$ is not a field I know definitions of ...
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42 views

Example of isomorphism despite different orders?

Is it possible to have an isomorphism between two groups even though they have different orders (specifically finite order)? How about an infinite order group and a finite order group? I'm asking ...
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40 views

Number of subgroups of $G$ conjugate to $H$

I need help in understanding following problem. Let $G$ be a finite group and $H\leq G$. Prove that the number of subgroups of $G$ conjugate to $H$ is a divisor of $|G|$. I want to understand what ...
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prove that group of order 275 has non trivial center.

Let $G$ be finite group of order $275 = 5^2\cdot11$. prove that $Z(G)=\{g\in G:\forall h\in G\space\space gh=hg\}\not=\{e\}$. Using the Sylow theorems I manged to prove that $G$ has normal subgroup ...
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General Classification of finite simple ternary groups?

Define a ternary group as an algebraic set endowed with a 3-ary operation f: that maps 3 elements onto another in the set. Furthermore for any three elements a,b,c there exists a unique 4th element ...
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1answer
82 views

Finite groups with nontrivial outer automorphisms

Let $G$ be a finite group whose center $Z(G)$ is trivial. Suppose that the group $\text{Out}(G)$ of outer automorphisms is nontrivial. Question: Does there always exist an $f \in \text{Aut}(G)$ ...
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Order of Aut$(D_4)$

How can I prove that order of Aut$(D_4)$ is 8. Let we show $D_4$ as $\{e,\sigma,\sigma^2,\sigma^3,\tau,\tau\sigma,\tau\sigma^2,\tau\sigma^3\}$ and ...
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1answer
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Reducibility of Cyclic groups

Let $G$ be the cyclic group $C_{4}$ and consider the 2-dimensional representations of G. Why does extending scalars to the complex numbers let this representation become reducible? I understand how it ...
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1answer
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$G=HK$ then the index of a subgroup is determined by $H$ and $K$

Let $G=HK$ s.t. $H\cap K=1$ and let $R$ be a any subgroup of $G$. I wonder necessary and suffucient condition for the equality, $$|G:R|=|H:H\cap R||K:K\cap R|$$ Note that if $H$ and $K$ are normal ...
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How is the entire $SO(2)$ group the standard rotation matrix?

In a book I am using, the following is presented, $$\mathcal{R}(\phi) = \begin{pmatrix} \cos (\phi ) &\sin (\phi ) \\ -\sin (\phi ) &\cos (\phi )\end{pmatrix}$$ The group's name is ...
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Independent components of a group cocycle

Fix a finite group $G$. An $n$-cochain of $G$ with coefficients in a $G$-module $M$ is a function$$b:G^n\rightarrow M$$To determine $b$, one must specify the values $b(g_1,\ldots,g_n)\in M$ for all ...
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1answer
43 views

Number of cyclic subgroups order $p^2$ in $\mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2}$

Let $$G={ {<a>}_{p} \times {<b>}_{p} \times {<c>}_{p^2}} \cong \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2} \text{, $p$ is prime}$$ There are $p^3-1$ elements with order ...
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Can this lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$. Let the lattice $\mathcal{L}$ as follows: ...
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1answer
25 views

Subgroups of group of characters of a finite abelian group

Let G be a finite abelian group, H a subgroup of the group of characters of G. Is it true that H is the group of characters of some quotient group of G? Thanks for any help.
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1answer
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$p^nm$ group, element of order $m$

Let $p$ be an abelian group of order $p^nm$, $p$ prime, and $p$ does not divide $m$. Is it true that the group must contain an element of order $m$, or a multiple of $m$? If yes, how to prove it? If ...
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Show that the group of permutations of {1,2,3,4} is equal to the product of it's subgroups…

Show that the group of permutations of {1,2,3,4} $$\sigma_4$$ is equal to the product of it's subgroups $$C_2\times C_2 $$ and$$D_6=(x^3=y^2=1, yx=x^2y)$$ I'm not sure whether to just multiply the ...
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1answer
24 views

Finding unique groups of C11 semidirect product C5

Find unique groups of form $$ C_{11}\rtimes C_5$$ (semi-direct product) with homomorphism $$h:C_5\rightarrow Aut(C_{11})$$ I've found the possible homomorphisms i.e $$h=Id,x^3,x^4,x^5,x^9 $$ So ...