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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2
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1answer
34 views

Proper subgroup of $S_{15}$ that strictly contains $\sigma $

How does one prove that: There exists no cyclic proper subgroup of $S_{15}$ that strictly contains the following permutation (of order $10$)? $$\sigma = (1, 2,3, 4,5, 6, ...
0
votes
0answers
12 views

Proofs of congruence relations

Exercise 2.3 from "Introduction to Mathematical Cryptography" Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$ of order $r$. (a) Suppose that $x = a$ and $x = b$ are both integer ...
5
votes
1answer
78 views

How to show $\frac{\mathbb{Z}_m\times \mathbb{Z}_n}{\langle (a,b)\rangle}\simeq \mathbb{Z}_{\frac mc}\times \mathbb{Z}_{\frac nd}$?

Suppose we have the group $\mathbb{Z}_m\times\mathbb{Z}_n$, and $(a,b)\in\mathbb{Z}_m\times\mathbb{Z}_n$. We need to justify that (i) There exist $c, d$ such that $\langle (a,b)\rangle$ is ...
1
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1answer
31 views

The set of isomorphisms from a right coset of the automorphism group $Aut(X)$ in $S_n$.

From "Lecture Notes in Computer Science" by Christoph M. Hoffmann , on page 22- Theorem 4 Let $X$ and $X'$ be two isomorphic graphs with vertex set $V = \{1 ..... n\}$ , Then the set of ...
0
votes
6answers
69 views

Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n

The question is given in the title. I am unable to come up with a counterexample and I'm thinking this could apply to cyclic groups but not necessarily to general groups. Does anyone have any ideas or ...
-1
votes
3answers
123 views

Only two groups of order $10$: $C_{10}$ and $D_{10}$

Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
0
votes
0answers
6 views

Generating space representation of all elements of a point group using generators

A well known set of groups are the 32 three-dimensional crystallographic point groups which elements represent the transformation matrices of the symmetry elements (rotation, reflection etc.). If one ...
0
votes
1answer
22 views

$G$ is a finite group, if $ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$

$G$ is a finite group of order $n$, then if $a,b\in G : ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$. multiply both sides by $ab^{-1}$ we get $(ab^{-1})^2 = ab^{-1}ab^{-1}=ba^{-1}ab^{-1}=1$ so ...
0
votes
0answers
17 views

Structure of frobenius groups.

Definition- A groups $G$ is called Frobenius if it has a proper nontrivial subgroup $H$ such that $H \cap H^g=1\ \forall\ g\in G-H$. Do we have a structure or classification theorem for (finite) ...
3
votes
3answers
46 views

Find all homomorphisms $Q \rightarrow \mathbb{Z}_8$

Let $Q$ be the quaternion group. Find all homomorphisms $\phi: Q \rightarrow \mathbb{Z}_8$ What I get into is one big ifology: Of course $\phi(1) = 0$, then $0 = \phi(1) = \phi(-1 \cdot (-1)) ...
2
votes
1answer
21 views

Action via automorphism

I want to ask what does it mean to say a group $A$ acts on $N$ via automorphisms. It is a notion used in M.Isaacs book and I am not familiar with. I tried to find how it is defined but a scanned e ...
26
votes
1answer
384 views

Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : the number of groups with order $n$. Can $N(n)=n$ hold for some $n>1$ ? I checked the OEIS-sequence as well as the squarefree numbers $n$ in the range $[2,10^6]$ and found no ...
1
vote
2answers
44 views

Showing a non-isomorphism of groups

I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$. $\Bbb Z^*_n$ means integers up to $n$ coprime with $n$ I do not know how to do this. I have difficulties doing proofs ...
1
vote
1answer
190 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 $\mathbb Z_{10}$ ...
0
votes
1answer
83 views

Number of non-isomorphic groups of order 21

Let $G$ be a group of order $21$. Find the number of non-isomorphic groups of order $21$ My solution: If the group $G$ is commutative,then $G$ can be expressed as a direct product of cyclic groups ...
1
vote
1answer
16 views

Finding all permutations which satisfy given condition

In a symmetric group $S_n$ find number of permutations $P$ such that in the disjoint cycle decomposition of $P$ , length of cycle containing $1$ is $k$ . Here's my attempt at this . I found number of ...
4
votes
0answers
48 views
+50

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
2
votes
3answers
1k views

How to determine the number of isomorphism types of groups of a given order?

if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ? for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ " Klein 4-group" ...
3
votes
1answer
75 views

Are there cube-free numbers $n$, for which the number of groups of order $n$ is unknown?

For squarefree $n$, there is a formula allowing to compute the number of groups of order $n$. I do not think that such a formula exists for cubefree numbers. If a cubefree number $n$ has the ...
4
votes
1answer
137 views

Is $n=8{,}574{,}796{,}230$ the smallest squarefree number $n$ with $gnu(n)>10^6$?

The number of groups of order $n$ (gnu(n)) can be calculated by a closed formula, if $n$ is squarefree. I could calculate the values with GAP, additionally, I programmed a version in PARI/GP, working ...
-2
votes
1answer
59 views

Is the symmetric group $S_4$ cyclic

Is the symmetric group $S_4$ cyclic? By writing all $24$ elements we can write the tabular form of $S_4$. Then choosing each element of $S_4$, we can find its order and thus, we can show that ...
1
vote
2answers
27 views

Which one of the below options is correct?

I think the option $(Q)$ is true since $O(Q/\{-1,1\})= 8/2 = 4 = 2^2$. Since order is $p^2$ thus $(Q)$ option is true. Can anyone suggest about option $(P)$? Thanks
1
vote
1answer
40 views

Prove that $g\in P$

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup. If $g$ is an element of $G$ with order a power of $2$ then it lies in some conjugate of $P$. I get this. But if also $g\in C_G(P)$, then it is ...
0
votes
1answer
31 views

Which of the following are isomorphic?

I am a beginning learner of group theory. Which of the following are isomorphic:$$\mathbb{Z_{24}}, D_{4}\times \mathbb{Z_{3}},A_{4}\times \mathbb{Z_{2}},\mathbb{Z_{2}}\times D_{6}, ...
3
votes
1answer
390 views

Groups of order $pq$ have a proper normal subgroup

I am doing the following exercise from [Birkhoff and MacLane, A survey of modern algebra]: Let $G$ be a group of order $pq$ ($p,q$ primes). Show that either $G$ is cyclic or contains an element ...
1
vote
1answer
39 views

Expressing $z\in G$ as $z=gh^2$ where $g$ is a $2$-element and $|h|$ is odd

Let $z\in G$ where $G$ is a finite group, then is it always true that there exist elements $g,h\in G$ such that $z=gh^2$ where $|g|=2^k$ for $k \in \Bbb{Z_{\ge0}}$ and $|h|$ is odd?
1
vote
0answers
24 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
6
votes
3answers
58 views

$H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\} , \forall g \in G \setminus H$ , then $|\cup gHg^{-1}|>\dfrac 12 |G|+1$

Let $H$ be a proper subgroup of finite group $G$ such that $H \cap gHg^{-1}=\{e\}$ for all $g \in G \setminus H$. Then is it true that $$|\cup_{g \in G \setminus H}gHg^{-1}|>\dfrac 12 |G|+1$$ If ...
0
votes
1answer
22 views

Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
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0answers
35 views
+50

If $M < M < G$ with certain conditions and a special subgroup $U < G$, then we can choose $|G / M| = p$ and $p \nmid |M|$.

Let $G$ be a finite group and $U \le G$ a subgroup of odd order. Assume that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t \notin U$. Also assume $U^g \ne U$ implies $U^g \cap U = ...
1
vote
1answer
32 views

Characteristic subgroups of order $2$

Could anybody give an example of a finite abelian $2$-group with more than one characteristic subgroup of order $2$ ? (In other words, a finite abelian $2$-group $G$ with a Klein subgroup $V$ such ...
1
vote
2answers
14 views

Multiplying elements of a multiplicative cyclic group

This is about correct operation in a multiplicative cyclic group. Say we have the group $\mathbb Z_4^{\times} = \{1,2,3\}$, if we multiply the elements, say $3\cdot 2\cdot 2 $ we get $12 \mod 4 = 0$ ...
1
vote
0answers
25 views

Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
1
vote
3answers
57 views

$G$ is a finite group, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$.

Prove / disprove: Let $G$ be a finite group of order $n$, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$. I can't find a group that will satisfy this condition so I think it's ...
3
votes
1answer
44 views

Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
15
votes
1answer
4k views

Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
1
vote
1answer
18 views

Semidirect product when the action factors

Suppose I am forming a semidirect product of finite groups $G \rtimes_\theta A$. Here, $\theta : A \to \mathrm{Aut}(G)$ is a homomorphism. Now, suppose I notice that the homomorphism $\theta$ factors ...
7
votes
2answers
129 views
+300

How are simple groups the building blocks?

I know a bit about simple groups. A finite Abelian group is a (direct) product of finite cyclic groups. The simple finite Abelian groups are exactly $\mathbb{Z}_p$ for $p$ a prime. And so, I ...
1
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0answers
39 views

Why is $M_{21}\cong PSL_3(\mathbb{F}_4)$?

I have recently been reading about the Large Mathieu Groups in "The Finite Simple Groups" by Robert Wilson and I have not come across any explanation of this isomorphism as of yet. In this context, ...
3
votes
1answer
47 views

How can I check whether the group $[16,13]$ in GAP with $3$ generators can be generated by $2$ elements?

The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same ...
3
votes
1answer
51 views

Which non-abelian finite groups have the property that every subgroup is normal?

If $G$ is an abelian group, every subgroup $H$ of $G$ is normal. I searched for non-abelian finite groups $G$ , such that every subgroup is normal and GAP showed only the groups $G'\times Q_8$ , ...
1
vote
0answers
6 views

Alternative construction of the twisted group $^2 E_6 (q)$.

I am looking for the alternative construction of the twisted finite simple group $^2 E_6 (q)$ possibly avoiding the Lie theory. Especially I am interested in calculating its order. Maybe some of you ...
0
votes
0answers
36 views

How can I prove that $G$ is not a simple group. [duplicate]

I only have that $|G|=4n+2$ for $n \in \mathbb{N}$. And I have to prove that G is not a simple group.
1
vote
1answer
15 views

Why is the Young symmetrizer non-zero?

Suppose $\lambda$ is a partition of the natural number $n$ and $T$ is a standard Young Tableaux of shape $\lambda$. Let $$P_{\lambda}:=\lbrace g\in S_n:g\text{ preserves the rows of }T\rbrace$$ and ...
1
vote
2answers
35 views

$G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$?

Let $G$ be a group of order $pn$ , where $p$ is a prime and $p>n$ , then is it true that any subgroup of order $p$ is normal in $G$ ? ( I know that any subgroup of index smallest prime dividing ...
1
vote
0answers
33 views

Quotient of $S_4$ by a normal subgroup

Is this true? > Quotient of $S_4$ by a normal subgroup of order 4 is Abelian ? As the group is of order 4!/4=6, so it is either $S_3$ or $\mathbb Z_6$. how to decide? please help.
0
votes
0answers
20 views

Image of Hall subgroup is a Hall subgroup

Let $\pi$ be any set of prime numbers. $H \subset G$ is a Hall $\pi$-subgroup if no pime dividing the index $|G:H|$ lies in $\pi$ and every prime divisor of $|H|$ lies in $\pi$. Let $\varphi: G ...
2
votes
1answer
45 views

Is the definition given by the GAP-manual equivalent to the one given in the site? [closed]

Here http://groupprops.subwiki.org/wiki/Normalizer_of_a_subset_of_a_group the normalizer of a subset of a group is defined. GAP gives the following description of the Function IsNormal : 39.3.6 ...
1
vote
1answer
39 views

If $G = G_{\alpha} \rtimes R$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \cong C_2\times C_2, D_3$ or $D_4$

Suppose that $G$ is a transitive permutation group and let $R$ be a regular normal subgroup which is isomorphic to the non-abelian group of order $27$ and exponent $3$. Then $G = R \rtimes ...
2
votes
0answers
18 views

Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

I have asked this question in MathOverflow, but it received no answers, so I am posting it here. In my recent work I have been led to consider the following type of permutation factorizations. Let ...