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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32 views

if each $(i,j)\space:\space g_ig_j=g_jg_i$, then $G$ is abelian

Let $G$ be finite group. say that $a,b\in G$ hold that $(a,b)\in R\subseteq G\times G$ iff $\exists g\in G \space:\space gag^{-1}=b$ note that $R$ is an equivalence relation. let $g_1,...,g_n$ be the ...
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2answers
27 views

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true?

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true? $1.G$ has exactly $4$ subgroups upto isomorphism. $2.G$ is abelian. $3.G$ is isomorphiq to a ...
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0answers
15 views

Find the subgroups of A4

I had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even ...
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1answer
18 views

Let $S$ be a subset of $R$ such that we have associative relation $*$ defined on $S\times S\to S$ with some properties…

Let $S$ be a subset of $\\R$ such that we have associative relation $\\*\\$ defined on $S\times S\to S$ with $$a*b*a=b \hspace{0.5cm} \forall a,b\in S, \hspace{1.5cm} \exists e \hspace{0.5cm} ...
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1answer
24 views

$H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,…,n)$ [on hold]

For $n>2$, if $H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,...,n)$ , then (A) $H = S_n$ (B) $H$ is abelian (C) The index of $H$ in $S_n$ ...
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0answers
54 views

Wielandt's Exercise [on hold]

Wielandt, Exercise 5.2. Assume that the intransitive group $G$ has degree $n$ and minimal degree $n−1$. If no transitive constituent of $G$ has degree $1$, then they all are faithful and all except ...
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25 views

about groups of order p^2qr [on hold]

i need help to understend next theorem (page 148) : https://archive.org/stream/jstor-1986340/1986340#page/n11/mode/2up Is same true for groups of order $p^2q^2r$?
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1answer
34 views

Why in this sense this homomorphism is injective?

In this proof:enter link description here Page 19, it gives a construction of outer automorphism of $S_6$,it sends $S_6$ to Perm($S_6$/H)= $S_6$, and by the former proof, it is injective.However, it ...
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2answers
49 views

Group with $p+1$ Sylow $p$-subgroups

Given a group $G$ with $p+1$ Sylow $p$-subgroups, I've deduced that $R = P \cap P'$, where $P, P'$ are Sylow $p$-subgroups, has index $p$ in each of $P, P'$; and that all $p+1$ Sylow $p$-subgroups of ...
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30 views

finite solvable group with a certain property

Let $G$ be a finite solvable group and for each proper normal subgroup $N$ of $G$, $\frac{G}{G^{\prime}N}\cong \Bbb{Z}_p\times\Bbb{Z}_p$ or $\Bbb{Z}_{p^n}$, where $n\geq 1$, $p$ is a prime number ...
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1answer
36 views

properties on groups of order $p^2qr$

I read somewhere that if $|G|=p^2qr$, $H\subseteq G: |H|= p^2q$, $p>q>r$ primes, then if only $H$ is maximal subgroup, then $H$ is Abelian. Is this problem correct? Are there any same properties ...
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1answer
29 views

Homomorphic images of a group [on hold]

If we consider $Q_8$ i.e. the Quaternion Group,then how to find the homomorphic images of this group?
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1answer
14 views

Primary decomposition of $Z_{1001}$ as a group of multiplication

The question is asking for the primary decomposition of $Z_{1001}$ as an abelian group under multiplication. So I did the following. By Euler $\phi$ function, I count the number of integers ...
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1answer
10 views

Using Lattice Isomorphism Theorem

I am working on this for my algebra class and I am stuck at the very end. $\textbf{QUESTION:}$ Let $p$ be a prime and let $G$ be a group of order $p^\alpha$. Prove that $G$ has a subgroup of order ...
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1answer
47 views

No simple groups of order 9555: proof

While reading through crazyproject, I came across the following proof that there were no simple groups of order $9555$. However, I do not understand the step that says: "Moreover, since 7 does not ...
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1answer
26 views

About primary decomposition of $\mathbb{Z}_7$

I want to find the primary decomposition of $\mathbb{Z}_7^*$ under multiplication group action. So I know 7 is a prime number so it is a group of order 6. So possibilities are $\mathbb{Z}_7^*\cong ...
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1answer
45 views

Group of order 36

I follow a proof which show that any group of order $36$ has a normal Sylow subgroup. The author suppose that $G$ has no normal subgroups of order $9$ or $4$. Then $G$ won't have subgroups of order ...
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0answers
24 views

Describe all subgroups of $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$

Is it possible to give a general discription of all subgroups of the group $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$ of inner automorphisms of $\operatorname{GL}_2(\mathbb F_{p^n})$? ...
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1answer
23 views

Subgroups in $G$ of the form $gHg^{-1}$

Let $G$ be a group and $H$ be a subgroup of finite index. Prove that there is only a finite number of distinct subgroups in $G$ of the form $gHg^{-1}$ where $g$ belongs to $G$.
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1answer
23 views

Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...
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1answer
21 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
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1answer
29 views

Prove $f(x) = x * a$ is bijective (preferably using inverse)

I have this question that I am stuck at. Let $(G; * ; I)$ be a group and let 'a' in $G$. Let $f : G \rightarrow G$ be the function defined by $f(x) = x * a$ for all $x$ in $G$. Prove that $f$ is ...
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2answers
38 views

Is this group cyclic?

Is this group cyclic, and what would be its generator? where H = {a+b(sqrt(2)) element of R | a, b element of Z} I know that in order for a group to be cyclic if the generator equals the group, but ...
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1answer
33 views

Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
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0answers
16 views

Conjugacy classes of PSL(6,7)

I need the conjugacy class sizes of projective special linear group PSL(6,7). I couldn't find it by using GAP. Could someone find it?
4
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1answer
39 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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2answers
27 views

Question on order of elements in groups (subgroups)

I am a bit confused at the moment, but what can we say about the order of all elements in a finite (sub)group? Suppose we have a group $G$ such that $|G|=p^k$ for a prime $p$. Next let $H$ be a ...
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2answers
19 views

Part of simple proof of nontrivial center in p-group

I'm trying to understand the proof of a Burnside theorem (as stated in Beachy's Abstract Algebra p. 328): Let $p$ be prime number. The center of any $p$-group is nontrivial. Now, In the proof they ...
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1answer
27 views

Elements of $\operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^k$

Let $p > 2$ be a prime number and $n\ge 1$ an integer, and consider the group $G = \operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^n(p^{2n} - 1)$. Let us denote by $\operatorname{Inn}(G)$ (the ...
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0answers
9 views

Characterise all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$

The title pretty much explains everything. Is it possible to give an easy characterisation of all subgroups of $\operatorname{SL}_2(\mathbb F_{p^n})$?
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2answers
37 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
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1answer
22 views

Center of a Group and automorphisms

The center of a group, G is defined as: Z = {a|ag = ga for all g ∈ G}. Let $\phi$ be an automorphism of a finite group G to G. Show that for any a ∈ Z then $\phi(a) $∈ Z. Conclude that $\phi(Z) = Z$. ...
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1answer
25 views

A finite and stable part of a group is a subgroup

How to prove that a a finite and stable part H of a group G is necessarily a subgroup ? This is equivalent to proving that every element x of H has its inverse in H too :)
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0answers
22 views

Groups where there's always a “deformation retraction” homomorphism onto any subgroup.

Let $G$ be a finite group with the property that, for every subgroup $H$, there exists a homomorphism $f: G\to H$ such that $f(h)=h$ for all $h\in H$. What possible groups can $G$ be? If $P$ is a ...
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1answer
39 views

A Lagrangian group

I have a non-Lagrangian group $G$ of order $pq^3$, $Q$ a Sylow $q$-subgroup of G and a $H$ a subgroup of $Q$ with $|H|=q^2$. It is clear that $Q \subseteq N_G(H)$. I must prove that $G$ doesn't posses ...
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29 views

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime.

Let $a$ be an element of order $n$ in a group $G$. If $a^m$ has order $n$, then $m$ and $n$ are relatively prime. Assume $a^m$ has order $n$ and, $m$ and $n$ are not relatively prime. Then ...
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0answers
31 views

Cosets of $H$ in the group $S_4$

In the group $S_4$, let $H$ be the subgroup of those permutations which leave $4$ fixed: $$H=\{(1),(12),(13),(23),(123),(132)\}.$$ List all of the left and right cosets of $H$ in $S_4$.
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1answer
30 views

how to count the number of subgroups of order n with some constraints

Let $G$ be a group of order 24. How many subgroups of $G$ are there, given that there are exactly 8 elements of order 3 in $G$? Tried: since $24=2^3\times 3$, there must be a subgroup of order ...
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0answers
15 views

Find number of triangle and quadrangles in $AG_2(3)$

$\textbf{Question-}$ Show that in $AG_2(3)$ there are - i) $72$ triangles ii) $54$ quadrangles iii) $4$ triangles in each quadrangle iv) $3$ quadrangles containing a given triangle My try- I know ...
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1answer
16 views

Let $M,N$ be isomorphic as $\mathbf{Z}[G]$-mods, are they isomorphic as $\mathbf{Z}[H]$-mods, where $H<G$

So I have recently been looking at the isomorphisms of $\mathbf{Z}[G]$-mods ($G$ finite), and noticed that a couple of my examples saw them isomorphic as $\mathbf{Z}[H]$-mods also, where $H$ is a ...
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1answer
44 views

Quaternion^2 in the symmetric group

Which is the minimum $n$ such as $Q_8\times Q_8\cong H$ with $H<S_n$? I now that the minimum n such as $Q_8$ embeds in $S_n$ is 8, so $Q_8\times Q_8$ embeds in $S_8\times S_8< S_{16} $, but is ...
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3answers
42 views

Group isomorphisms and a possible trivial statement?

I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|} \hline \circ& a & b & e \\ \hline a& ...
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1answer
38 views

To show a finite group G is nilpotent

Let G be a finite group and G' denotes it's commutator.If order of G' is 2.Then show that G is Nilpotent. What I have tried:G/G' is abelian,so it is nilpotent again G' is nilpotent as it's order is ...
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0answers
21 views

Finding Semi-direct products of Z/3Z and Z/7Z

This problem originated from trying to find the group isomorphisms of groups of order 21. I already worked out that there's one subgroup of order 7 that's normal, and one subgroup of order 3. Even ...
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1answer
21 views

Order of groups and elements

(related to this question: Finite Group and normal Subgroup) Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ We define the set $X:= \{g\in G | g^d=1\}$ ...
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1answer
32 views

In a non abelian group of order $p{^4}$ Quotient of center by commutator is abelian

Let $G$ be a non abelian group of order $p{^4}$,$p$ is a prime.Let $N$ be a normal subgroup of $G$ with |$N$|=$p$ and $G/N$ is abelian.Then prove that $N$ is a subgroup of $Z(G)$ and $Z(G)$/$N$ is a ...
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1answer
62 views

Finite Group and normal Subgroup

Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ and define the set $X:= \{g\in G | g^d=1\}$. Show: if $H$ is a normal subgroup of $G$ with order d then ...
2
votes
3answers
72 views

Find order of group given by generators and relations

Let $G$ be the group defined by these relations on the generators $a$ and $b$: $\langle a, b; a^5, b^4, ab=ba^{-1}\rangle$. I need hints how to find order of $G$.
3
votes
1answer
68 views

Show that the order of $G$ cannot be divisible by $7$.

Show that the order of a proper primitive group of degree $19$ cannot be divisible by $7$. It means as following- This means that a group $G$ which acts on a set $\Omega$ of cardinality $19$ (this ...
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0answers
41 views

Can a chain subcomplex be a direct summand but it's 'origin' is not? (group homology)

Let $H\!\leq\!G$ be finite groups and $C_\ast(H)\!\leq\!C_\ast(G)$ their bar complexes (each $C_k(G)$ is a free $R$-module with basis $(G\!\setminus\!\{1\})^k$). Is it possible that $H$ is not a ...