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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20 views

Group presentation: How can we determine the group with the presentation below?

Please how can we determine the finite group whose presentation is given as: $$G:=\left\langle g,h|g^4=h^4=1,hg=g^{-1}h \right\rangle?$$
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0answers
19 views

History of Dihedral Groups?

I'm trying to write a brief historical outline for a project concerning Dihedral Groups, but It has been to hard for me to find information about the first time these groups appeared on literature. If ...
2
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1answer
27 views

If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?

Given $|G|=p^2$ then how can you deduce $G\cong C_{p^2}$ or $G \cong C_p \times C_p$ I have shown that G is abelian, not sure what to do next
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1answer
20 views

Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
2
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3answers
53 views

Let $G$ be a group of order 505.Is it cyclic?

Let $G$ be a group of order 505.Is it cyclic? My try:By Sylow's theorem $G$ has a subgroup of order 101 and a subgroup of order 5.If both have been unique we could have concluded that $G$ is cyclic. ...
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0answers
9 views

Irreducible character of degree greater than one takes value zero on some conjugacy class

It is a standard fact that irreducible character of a finite group of degree $>1$ takes value $0$ on some conjugacy class. A proof for example can be found here. I would like to know whether there ...
5
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1answer
54 views

Every group as full symmetry group of points in $\mathbb R^d$

Does every finite group $G$ have the property that it is isomorphic to a full symmetry group of some set of points in $\mathbb R^n$ for some $n$
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1answer
40 views

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 ...
4
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1answer
63 views

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 ...
1
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1answer
49 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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1answer
31 views

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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0answers
29 views

The complexity of finding the order of an element in a finite group, and some other problems [on hold]

I'm new to group theory. Let's focus on the finite group $Z_N$ where $N$ is a large number (possibly hundreds of digits). I want to discuss whether there exists an efficient algorithm to solve the ...
2
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1answer
30 views

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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0answers
20 views

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 ...
0
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1answer
18 views

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 ...
2
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1answer
34 views

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 ...
0
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2answers
45 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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0answers
43 views

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 ...
0
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0answers
24 views

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. ...
29
votes
2answers
999 views

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 ...
0
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3answers
29 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
47 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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1answer
30 views

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 ...
1
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2answers
69 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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0answers
21 views

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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0answers
23 views

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 ...
4
votes
1answer
38 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, ...
11
votes
1answer
93 views

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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2answers
37 views

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 ...
0
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1answer
34 views

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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0answers
39 views

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 ...
0
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1answer
37 views

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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0answers
37 views

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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0answers
9 views

$G$ 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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0answers
29 views

Permutation Group acts properly?

Consider the following definition of a proper group action: Let Γ be a group acting by isometries on a metric space X. The action is said to be proper if for each x ∈ X there exists a number r > 0 ...
0
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1answer
22 views

$|D_{18}|=18=2 \times 3^2$, 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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0answers
11 views

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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0answers
28 views

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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1answer
63 views

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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2answers
54 views

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 ...
4
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1answer
58 views

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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3answers
59 views

$G/Z(G)$ 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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0answers
13 views

Is quotient of Euclidean space by action of permutation group a Hadamard space?

Suppose that a permutation group G acts on some Euclidean space E. Is the resulting quotient space with induced metric a Hadamard space (possibly after completion)? If not, what is missing?
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0answers
12 views

calculate of maximal normal $\pi$-subgroup of $G$

If $\pi$ is a set of primes, then a $\pi$-number is an integer n all of whose prime factors lie in $\pi$; the complement of $\pi$ is denote by $\pi'$. A group $G$ is a $\pi$-group if the order of each ...
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2answers
44 views

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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0answers
26 views

How do different group extensions behave?

The concepts of split extensions, non-split extensions, and semi-direct products are new to me. I am aware of the definitions (regarding short exact sequences), but I am unsure how to work with such ...
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2answers
37 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 ...
1
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2answers
37 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 ...
0
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2answers
42 views

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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0answers
24 views

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 ...