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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14 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language
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1answer
35 views

Quantity of elements of order $d$ in $Z_n$, with $d | n $

I'm studying for an exam and I can't answer this problem. I'd appreciate a hint. What I've got so far: Let $x$ be an element or order $d$. Then, $x\cdot d \equiv 0 \pmod{n} \Rightarrow n \mid x\cdot ...
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0answers
45 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
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1answer
27 views

Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
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109 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and ...
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11 views

Maximal subgroups of $PSL(n,q)$

We can match $PSL(2,2)$ with $S_3$, $PSL(2,3)$ with $A_4$, and so on. Also $PSL(n,q)$ is simple in the exception of the two listed. What are the maximal subgroups of $PSL(n,q)$?
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1answer
26 views

Latin square property sufficient?

So I know that for any group table, Every row must contain distinct group elements and the same holds for every column for a group table. And this property is called the Latin Square property. ...
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37 views

Am I correct regarding Aut($Z_n$)

In the following pic- shouldn't it be $\Bbb{Z}_{{p_j}-1}$ instead of $\Bbb{Z}_{p_j}$. I think so because Aut$(Z_{p^n}) \cong Z_{p-1} \oplus \underbrace {(Z_p\oplus Z_p \oplus \dots Z_p)}_{n-1\ ...
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1answer
31 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
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2answers
48 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
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51 views

Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
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1answer
34 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
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38 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
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1answer
32 views

Why can we assume $N$ to be a $p$- group?

Let $G$ be a finite solvable group such that if three distinct primes $p,q$ and $r$ divides $|G|$ then $G$ does not contain any element of order the product of two primes and $G$ is minimal w.r.t ...
2
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1answer
43 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
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3answers
71 views

Find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $.

I'd like to find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $. What I've tried so far: I tried to do $ \varphi(Id) = \bar{0} = \bar{4} $ (as somebody used here). But then I realized ...
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1answer
26 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with ...
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0answers
18 views

One dimensional representations of $SL_2(\mathbb{Z}/n\mathbb{Z})$

Someone knows a reference or knows how to calculate the linear character of $SL_2(\mathbb{Z}/n\mathbb{Z})$, for an arbitrary $n$? Thanks
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0answers
41 views

Factorizations of Finite Abelian Groups

Every finite abelian group $G$ can be uniquely written as $$\mathbb{Z}/{d_1\mathbb{Z}} \times \mathbb{Z}/{d_2\mathbb{Z}} \times \cdots \times \mathbb{Z}/{d_r\mathbb{Z}},$$ where $d_i$ divides ...
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1answer
52 views

Upper bounds for the number of intermediate subgroups

Assume that $G$ is a finite group, and $H\le G$ a subgroup of index $n>1$. What can we say about the number of distinct intermediate subgroups $K$, i.e. groups such that $H\subset K\subset G$? ...
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51 views

Permutation of cosets

Let $G$ be a finite group and $\gamma \in Sym(G)$, such that $\gamma (1) = 1$ and $\gamma (gH) = \gamma (g)H$ for all $g\in G$, $H\leq G$. This means $\gamma$ induces a permutation of the left cosets ...
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3answers
75 views

Prove that $|GL_n(\mathbb{F})|< q^{n^2}$.

Let $\Bbb F$ be a finite field, say $|\Bbb F|=q$; then we know that $|GL_n(\Bbb F)| < \infty$. But how can we prove that $|GL_n(\mathbb{F})|< q^{n^2}$? I'm guessing because there $n^2$ ...
4
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1answer
58 views

Generating series - Finite groups of order $n$

I am wondering if something of interest can be said about one of the two series $$G_1(x)=\sum_{n=1}^{+\infty}{\mathcal{G}(n)z^n}$$ $$G_2(s)=\sum_{n=1}^{+\infty}{\frac{\mathcal{G}(n)}{n^s}}$$ where ...
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35 views

Are there groups of order $p^4q^2$ which are not semi-direct product?

It is easy to show that if $G$ is a group of order $p^2q^2$, where $p,q$ are primes with correspondings Sylow subgroups $P,Q$, that $G$ is a semi-direct product of $P$ and $Q$. Moreover, if $pq\neq ...
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2answers
84 views

Classifying groups of order $5 \cdot 11 \cdot 61$

My question is whether I'm classifying groups of order $5 \cdot 11 \cdot 61$ correctly. (This is a qualifying exam question, so I also want to make sure that I'm doing it “efficiently”.) Sylow's ...
22
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1answer
792 views

Rotman's exercise 2.8 “$S_n$ cannot be imbedded in $A_{n+1}$”

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid ...
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1answer
24 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to ...
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1answer
13 views

Existence of a special Central cyclic subgroup

I have two related questions. Let $G$ be a finite $p$ group. Can we always choose a central subgroup $N$ of order $p$ not contained in the commutator subgroup? Clearly we cannot do that for extra ...
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1answer
28 views

Abelianization of a $p$ group.

Let $G$ be a $d$ generated finite $p$ group. Let $N$ be normal subgroup of order $p$ contained in $[G,G]\cap Z(G)$. Can we say say that $(G/N)/[G/N,G/N]=(G/N)_{ab}$ is a direct product of $d$ cyclic ...
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34 views

Covering groups

I am studying the Steiner system $S(5,6,12)$ and the ternary extended Golay code $\mathscr{C}_{12}$. The automorphism group of the Steiner system is the Mathieu group on twelve elements $M_{12}$ ...
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1answer
59 views

On cyclic decomposition of element in $S_n$

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...
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1answer
137 views

Which of the following abelian groups are cyclic groups?

Given the abelian groups of order $7425$: $$Z_{33} \times Z_{15} \times Z_{15} , \ Z_{25} \times Z_{297} , \ Z_{45} \times Z_{165} , Z_{55}\times Z_9 \times Z_{15}$$ Which of these groups, if ...
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1answer
38 views

Fulton and Harris: Exercise 1.3 in section 1.1

This is exercise 1.3 on page 5 of Fulton and Harris Representation Theory: A First Course. Exercise: Let $G$ be a finite group, let $V$ be an $n$-dimensional $\mathbb C$-vector space and let $\rho: ...
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63 views

Prove that $|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$ [closed]

Suppose that $G$ is a group of order $n > 1$, prove that $$|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$$ where $k=\lfloor \log_2 (n-1)\rfloor$.
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1answer
36 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
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33 views

Solving the Character table for $A_4$ and derived algebra characteristics.

Say I am given a group with 4 conjugacy classes $C_1, C_2, C_3, C_4$ with orders 1,3,4,4 respectively. I will call the conjugacy class characters $\chi_1,\chi_2,\chi_3,\chi_4$ respectively. I am ...
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22 views

What's the asymptotic of the radius of the Rubik square Cayley graph?

This post is a sequel of The Rubik Square permutation groups, which should be read first to understand the notation. Question: what's the radius$^*$ of the Cayley graph of $G_n$ generated by the red ...
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1answer
67 views

The Rubik Square permutation groups

This post was inspired by this webpage of mathematical challenge due to Mickaël Launay (French). Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as for the following ...
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1answer
84 views

Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?

I saw this question on an old qualifying exam: Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$? A simpler problem would be to show ...
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1answer
62 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
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2answers
81 views

How to obtain a presentation for each group of order $64$

I am new to his forum, and would like to know how to obtain a presentation for each group of order $64$. I wish to do this for all the groups of order $64$. Thanks in advance.
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3answers
74 views

Finding the number of elements of particular order in the symmetric group

I know how to find the order of element in any group $G$, for example the order of $2$ in $\mathbb{Z}_5$ is $5$ as $2 + 2 + 2 + 2 + 2 = [10]_5 = 0 0$, which is the identity in $\mathbb{Z}_5$. But, ...
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2answers
61 views

Why is $U(10)\not\approx U(12)$?

I'm trying to understand this example on isomorphism but failing to do so. I know that if two groups have a differing number of elements of each order they are not isomorphic. I assume this is what ...
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19 views

Matrices of Quotient Representation

I've been working through some exercises in Sagan's book on the representation theory of the symmetric group and I came across the following exercise that's been giving me some trouble: Let $X$ be a ...
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2answers
37 views

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. [duplicate]

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. Prove that, $\exists$ $x\neq e$ such that $x^2=e$ where $e$ represents the identity of $G$.
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2answers
83 views

Irreducible representations of $\operatorname{GL}_3(\mathbb{F}_q)$

I am trying to find all irreducible representations of $G = \operatorname{GL}_3(\mathbb{F}_q)$. I know that the order of $G$ is $(q^3-1)(q^3 - q)(q^3 - q^2)$ and the number of conjugacy classes is ...
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46 views

Reference request: groups of order $p^4$.

I am looking for a textbook or a paper which include the classification of groups of order $p^4$ ($p$ is prime) using generators and relations. In particular I like to understand which group $G$ ...
2
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1answer
38 views

Intuition on Hall subgroups and solvability

I see there are many questions on Hall subgroups, but I can't find one that answers my question. Let $G$ be a finite group. A subgroup $H<G$ is a Hall $\pi$-subgroup if its order is the product of ...
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4answers
365 views

How can I show that $A_5 = \langle (1,2,3), (1,2,3,4,5) \rangle$?

Show $A_5 = \langle (1,2,3), (1,2,3,4,5) \rangle$. Guys how can I show this in the best way.
3
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0answers
44 views

Schur multiplier of “large” groups.

Let $G$ be a finite group and let $M(G)=H^2(G,\mathbb{C}^*)$ be its Schur multiplier. For "small" groups I can compute the Schur multiplier by hand in terms of corresponding roots of unity. However, ...