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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100 views

What differences are there between $\mathbb Z_p$ and $\mathbb F_p$?

I read some books about finite fields, sometimes the author refers to the finite field $\mathbb{F}_p$ and sometimes to the finite cyclic group $\mathbb Z_p$. What is the difference between them?
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1answer
90 views

On $GL_2(\mathbb F_3)$

Consider $G:=GL_2(\mathbb F_3)$. I have to extrapolate as much information about it as I can. Without computations. First of all: I think someone else has already done this before me, hence if you ...
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1answer
98 views

How many elements can we obtain by that way?

Let $G$ be a finite group with $n$ elements with initial order $g_1,g_2,...,g_n$ and let's create the group multiplication table of $G$ with this initial order. In that table you will have $n$ rows ...
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1answer
164 views

How to tell whether a representation of a group is faithful or unfaithful?

From just the character table and the basis functions of the irreducible representations, how do I know whether a representation is faithful or unfaithful? For the 1-D representation it is trivial to ...
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1answer
49 views

a step in a proof in Samuel's Algebraic number theory

In the proof of Dirichlet's unit theorem, in Algebraic number theory by Samuel, there is a step in the proof that i don't understand. (p.73 in the french edition). He first introduces the logarithmic ...
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1answer
120 views

isomorphism between group and product of kernel by image [duplicate]

If $\phi$ is a morphism between groups $G$ and $H$, is $G$ isomorphic to $$ker(\phi)\times im(\phi)$$ ? Why ? Thanks.
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1answer
70 views

Orbit-Stabiliser Theorem applied to Symmetric group S_n

Let $G$ be the symmetric group $S_n$ acting on the n points $\{1,2,...,n\}$, let $g \in S_n$ be the n-cycle $(1,2,3,....,n)$. By applying the Orbit-Stabiliser Theorem or otherwise, prove that ...
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2answers
49 views

A sufficient condition for a finite group to have a $p$-power order

Let $G$ be a finite group and $p$ a prime number. If $G$ is a $p$-group, i.e. the order of every element of $G$ is a power of $p$ then is the order of $G$ equal to some power of $p$? How do I show ...
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1answer
23 views

Minimal Frattini Extensions

From "Construction of Finite Groups" article : "To construct minimal Frattini extensions of $H$ we have to consider all suitable irreducible $H$-modules $M$. Clearly, $M$ can be viewed as an ...
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0answers
53 views

Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
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1answer
90 views

Question about Sylow $p$-subgroups

If a group $H$ has order $255$ then the Sylow theorems tell us that it must have a Sylow $p$-subgroup of order $5$ and there are either $1$ or $51$ of them, also there is either $1$ Sylow $p$-subgroup ...
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1answer
175 views

Constructing groups with a given subgroup

I have a finite group $H$ and a number $n$ and would like to construct all groups $G$ of order $n$ such that $H$ is a subgroup of $G$. (In fact, I would prefer to construct only those which have ...
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1answer
47 views

Group of numbers in residue number system of first $n$ primes

Denote $$\Pi_n = \Pi_{i=1}^n p_i$$ ie. the product of the first $n$ primes. Assume we have a number $m$. Denote $$L(m) = k \iff \Pi_{k-1} < m \le \Pi_{k}.$$ Assume also $L(1)=1.$ Now, if $L(m) = ...
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3answers
249 views

Can a group with exactly five subgroups be nonabelian?

I was wondering if there is an example of a nonabelian group $G$ with exactly five subgroups. Let $G$ be a such group, and let $a,b\in G$ be such that $ab\ne ba$. Let us concentrate on the subgroups ...
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0answers
74 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...
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2answers
68 views

Formula linking size of centralizer and number of conjugacy classes for a finite group $G$

My course says it's easily explained that $\sum_{\substack{g\in G}} |C_G(g)|=m\times|G|$ where $m$ is the number of conjugacy classes of $G$. I don't think I see it that easily... Can you tell me ...
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1answer
67 views

Lemma about extra special group of order $p^3$

I am trying to understand the proof of the following lemma: Assume $P$ is a nonabelian group of order $p^3$ where $p$ is an odd prime. Assume also that $P$ has exponent $p^2$. Then ...
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1answer
135 views

angles between simple roots are obtuse, problem with proof

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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3answers
43 views

Question about normal subgroups in finite groups

I want to show that if $G$ is a finite group and $H$ is normal in G, and $K$ is a subgroup of $G$, and $H\cap K = \{e\}$ then $|HK|=|H||K|$
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2answers
71 views

Why is there no group of order 16 with 12 elements of order 8?

Let $|G|=16$ and $G$ contain 12 elements of order 8. I want to show that no such group can exist. (I've verified with GAP and on groupprops that there is no such group, I want to know why.) I have ...
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2answers
26 views

Automorphism group of $C_{p^2}$

I have a question about the automorphism group of the cyclic group of order $p^2$ where $p$ is a prime. In my notes, I have written that this group has a normal Sylow $p$-subgroup (i.e. only one Sylow ...
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0answers
30 views

centre of a p group [duplicate]

How do you show that a group of prime power has a non-trivial centre? I keep seeing this over and over but I can't solve it. I think it has something to do with the orbit stabiliser theorem and action ...
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1answer
131 views

Finite $p$-group in which all its maximal subgroups are cyclic

Let $G$ be a finite $p$-group, $|G|=p^n$. Let $M_1,\dots,M_r$ be all the maximal subgroups and suppose they are cyclic. Why is $\Phi(G)\le Z(G)$? $\Phi(G)$ is the Frattini subgroup. I have no idea ...
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1answer
93 views

On $p$-groups with a unique minimal subgroup

If $G$ is a finite group with a unique minimal subgroup, we know that $|G|=p^n$. I have to prove that if $p\neq2$ then $G$ is cyclic. This is the contest. What I don't understand is the following ...
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1answer
115 views

$MN/M \cap N \cong (MN/M) \times (MN/ N )$

I want to prove the following exercise from Dummit & Foote's Abstract Algebra: Let $M$ and $N$ be normal subgroups of $G$ such that $G=MN$. Prove that $G/M \cap N \cong (G/M) \times (G/N).$ ...
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1answer
123 views

Finite groups with a certain Frattini subgroup

Let $G$ be a finite group different from a cyclic $p-$group and $\Phi(G)=M_i\cap M_j$, where $M_i$ and $M_j$ are two arbitrary distinct maximal subgroups of $G$ and $i, j \geq 1$. Is it possible to ...
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4answers
103 views

$SL(2, \mathbb F_3)$ does not have a subgroup of order $12$

Using the characteristic polynomial I can prove that $SL(2, \mathbb F_3)$ does not has an element of order $12$, but how can I prove that $SL(2, \mathbb F_3)$ does not has a subgroup of order $12$?
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3answers
43 views

Is it true that $a$ and $b$ should be disjoint permutations?

Let $a,b \in S_n$ and $ab=ba$ and $b$ moves some points that not moved by $a$. Is it true that $a$ and $b$ should be disjoint permutations? EDIT: We can consider $b=a^kc$ where $a$ and $c$ are ...
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1answer
91 views

Proof that $A_4$ is the unique group of order $12$ with no subgroup of order six

Is there a simple proof that $A_4$ is the only group of order $12$ containing a subgroup of order six? (i.e. if $G$ is a group of order $12$ not having a subgroup of order six, then $G \cong A_4$?)
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1answer
125 views

Angle between roots in a root system

Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = ...
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2answers
190 views

a group with specific orders of elements

I want to find a group with elements of order $1,2,3,4$ and $5$ (at least one of each order). All I can say is that the order of the group is $60$ itself, but cannot find the correct one. Please let ...
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2answers
62 views

Is there a 20-order abelian subgroup of $S_5$?

The title says it all: Is there an abelian subgroup of order 20 of $S_5$, the group of permutations of five elements? Thanks for reading!
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1answer
56 views

every principal factor of a finite soluble group is elementary abelian.

every principal factor of a finite soluble group is elementary abelian. I am a little confused in a lot of definitions and I stuck in this exercise,this is so great if you just give me hints that I ...
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1answer
46 views

suppose $G$ is group,if $Aut(G)$ is abelian then $G$ is nilpotent from nilpotency class of at most 2.

suppose $G$ is group,if $Aut(G)$ is abelian then $G$ is nilpotent from nilpotency class of at most 2. it will be great if you help me how should I prove this.any note or reference will be great.thank ...
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2answers
179 views

On the numbers of maximal subgroups of a $p$-group

Let $G$ be a $p$-group, i.e. $|G|=p^n$. Call $\Phi(G)$ the Frattini group of $G$. Then we have that $G/\Phi(G)\simeq(C_p)^d$ ($d$ copies of the cyclic group of order $p$, i.e. ...
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1answer
45 views

$G'G^p=\Phi(G)$

Given a $p$-group $|G|=p^n$, consider $G'=[G,G]$ and define $G^p:=\langle g^p\;:\;g\in G\rangle$: then we have that $G'G^p=\Phi(G)$, where $\Phi(G)$ is the Frattini subgroup, defined as the ...
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1answer
51 views

Conjugacy between subgroup and group

Let $H$ be a proper subgroup of $G$, a finite group. Show that there exists $x\in G$ which is not conjugate to an element of $H$. Attempt: let $G$ act on $X$, the set of all proper groups of ...
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3answers
111 views

How can I prove that $Aut(C_p\times C_p)\simeq GL_2(\mathbb Z/p\mathbb Z)$?

How can I prove that $Aut(C_p\times C_p)\simeq GL_2(\mathbb Z/p\mathbb Z)$? No theorical argument came to my mind, so I'm trying to build explicitly an isomorphism $\phi:Aut(C_p\times ...
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1answer
55 views

Finding the hypercenter of a finite group in GAP

I usually find the hypercenter of a finite group by the command Hypercenter:=Union(UpperCentralSeries(g)); Its look odd, since I take the union of all the $i$th center of $g$ and "Hypercenter" is ...
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2answers
455 views

Is there any clasification of such groups?

Let $G$ be a finite group such that for all $x\neq e$, $$C_G(x)=\langle x\rangle$$ Is there any classification of such groups ?
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1answer
40 views

Detail in Lemma 6.13 pag. 190 from Isaacs Book

I'm all right in the proof till the point in which he says: the group $\langle a^2\rangle$ can contain no element of order $4$. First: what does "CAN CONTAIN NO ELEMENT etc" mean exactly? English is ...
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1answer
41 views

Extension of a group by a module

What is the definition of an extension of a group by a module? and what is the difference between extensions by groups and modules?
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150 views

A question from Isaac's Book

Let $A$ act via automorphisms on $G$, and $(|A|,|G|)=1$ and $G=HK$ where $H,K$ are $A$ invariant subgroups of $G$. Show that $$C_G(A)=C_H(A)C_K(A)$$ I can solve the question when I assume $H\cap K ...
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2answers
123 views

The periodic nature of the fibonacci sequence modulo $m$

Let $x_n$ denote the $n$-th element of the fibonacci sequence and $$A:=\begin{pmatrix} 0&1\\1&1 \end{pmatrix}$$ It's easy to show, that it holds: $$A^n=\begin{pmatrix} ...
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1answer
97 views

Unique minimal subgroup

We are in the contest of the classification of all groups of order $2^3$. We know that a group $G$ with a unique maximal subgroup is necessarely cyclic. 1)Then my teacher said that the dual ...
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1answer
144 views

Can finite non-isomorphic groups of the same order have isomorphic endomorphism monoids?

This is related to If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$? and Can non-isomorphic abelian groups have isomorphic ...
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2answers
41 views

Can someone give an example when this (particular construction of a ) normal subgroup is not characteristic?

Suppose $G$ is a group, and that in the action of $G$ on itself by right multiplication, we find an element $g$ of odd order. This tells us that $G$ intersects nontrivial with $A_{|G|}$ when we embed ...
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2answers
73 views

Determine the isomorphism class of M/T(M)

Let $M=\Bbb{Z}\oplus\Bbb{Z}\oplus\Bbb{Z}$ and $T: M\rightarrow M$ given by $T(x,y,z)=(4x+2z,2y,2x+10z)$. Show the cokernel $M/T(M)$ is an abelian group of order $72$, and determine its isomorphism ...
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2answers
179 views

How many nonabelian groups of order 2009? (Check work)

I just need someone to check this argument. Let $G$ be a nonabelian group of order $2009$. The prime factorization of $2009$ is $7^2 \cdot 41$. Let $n$ be the number of Sylow 7-subgroups. Then $n ...
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3answers
82 views

Is it possible that $N_G(H)=H$ and $N_G(K)=K$ where $K\subsetneq H$?

Is it possible that $N_G(H)=H$ and $N_G(K)=K$ where $K \subsetneq H$ and $H,K$ are proper subgroups of $G$ ?