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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$P\in Syl_p(G)\Rightarrow|G:N_G(P)|=|Syl_p(G)|$

Let $G$ be a finite group s.t. $|G|=p^rm$, where $(p,m)=1$. Let then $P$ be a $p$-Sylow subgroup of $G$, i.e. $P\le G$ with $|P|=p^r$. We want to show that $|G:N_G(P)|$ is the number of $p$-Sylow ...
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1answer
35 views

Subgroups of $\mathrm{PSL}(2,q)$ of order $2q$

Let $q\equiv 1\pmod 4$. Is it true that $\mathrm{PSL}(2,q)$ has a unique class of conjugate subgroups of order $2q$? I looked at the references that appear in this MO question, the only relevant ...
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3answers
66 views

Which of these groups are isomorphic?

A computer program outputs the following Cayley tables for groups of order 4. Wikipedia tells me that there are only two groups of order 4, the cylic group ($Z_4$) and the Klein four-group ($Dih_2 = ...
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1answer
74 views

Question about $p$-Sylow subgroups of the quotient group

I have been working on the following problem. Let $G$ be a finite group, $N\trianglelefteq G$ and $p$ a prime, then $n_{p}(G/N)\leq n_{p}(G)$. I have beeen trying to solve it, but it seems I ...
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1answer
39 views

Exploring $GL_2(\mathbb F_3)$

Looking at Sylow questions on $GL_2(\mathbb F_3)$. we have that $Q$ is the unique $2$-Sylow of $N=SL_2(\mathbb F_3)$. $|Q|=8=2^3$ hence by the classification of groups of order $p^3$, we have 5 ...
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1answer
46 views

Sylow questions 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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2answers
43 views

Subgroup Lattices and Dimension

I apologize in advance in the case that this question is nonsensical. If the idea isn't clear, I can perhaps explain more below. In the fall I am taking an undergraduate abstract algebra course, and ...
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68 views

If Q is a p-Sylow-Group of H there is a p-Sylow-Group P of G with $\phi(P)=Q$ while $\phi:G\rightarrow H$ epimorphism

Let G be a finite group and $\phi: G \rightarrow H$ a group-epimorphism. Proof: If $Q\in Syl_p(H)$ there is a $P\in Syl_p(G)$ with $Q=\phi(P)$.
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1answer
36 views

A little help with $p$-group, normal series, trivial action on factor groups.

Here is what I assume: $P$ is a $p$-group and there is a series of subgroups $P_0\unlhd P_1\unlhd \cdots\unlhd P_m = P$ such that $P_i\unlhd P$ for each $0\le i\le m$ and $P_0\le \Phi(P)$. We also ...
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4answers
57 views

Help with groups

let $G$ be a finite group with $e$ Identity element and let $a$ and $b$ belong to $g$ prove that if: $\gcd(o(a),o(b)) =1$ then $\langle a \rangle \cap \langle b \rangle = \{ e \}$. if someone can ...
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1answer
58 views

On some subgroups of $GL_2(\mathbb F_3)$

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
74 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
51 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
93 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
52 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
36 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
33 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
34 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
41 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
19 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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31 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
54 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
95 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
30 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
212 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
27 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
32 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
49 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
87 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
33 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
53 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
23 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
25 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
59 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
40 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
39 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
60 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
81 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
40 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
63 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
44 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
141 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
53 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
36 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
19 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
66 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
37 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
43 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
67 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
31 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 ...