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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0
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1answer
9 views

About $ S $-free group and normal subgroup of $ S $

Let $ S $ be a group. A group $ G $ is called $ S $-free if no quotient group of any subgroup of $ G $ is isomorphic to $ S $. Let $ G $ is finite group that is $ S $-free. if $ N \lhd S $, then is $ ...
2
votes
0answers
16 views

On a class of groups of order $p^2q$

Let $|G|=p^2q$ with following conditions: Sylow-$p$ subgroup is normal and is $\langle x,y\rangle \cong \mathbb{Z}_p\times \mathbb{Z}_p$. Sylow-$q$ subgroup is $\langle z\rangle$, and is not normal. ...
1
vote
0answers
20 views

$ \Phi(G) = 1 $ or $ \Phi(G) \neq 1 $?

Let $ G $ is a finite group. Suppose $ H \unlhd G $ such that $ G/H $ is supersoluble. Suppose $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup of $ G $. Suppose ...
4
votes
0answers
33 views

Why $ K \cap H $ is a maximal subgroup of $H $?

Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal ...
1
vote
0answers
23 views

$K, N < G$ and $|G:N|$ and $|K|$ coprime, then $K<N$. Group action argument?

A common exercise in finite group theory is Suppose $G$ is a finite group with $K < G$, $N \lhd G$. Suppose that $|K|$ and $|G : N|$ are coprime. Then $K < N$. The normal way to attack ...
1
vote
1answer
49 views

$[G:H] < \infty$ then $gHg^{-1} = H$ and is it true that $gHg^{-1} = H$

G is a group. $H < G$ and $ g \in G$ $gHg^{-1} \subset H$ I need to prove the following : a) if $[G:H] < \infty$ then $gHg^{-1} = H$ b) without the additional fact given in (a) is it true ...
1
vote
0answers
16 views

Supersolubility of $ G/H $ and $ H $ not deduced supersolubility of $ G $.

I want show $ S_{4} $ isn't supersoluble group. For this suppose $ 1 \leq B_{4} \leq A_{4} \leq S_{4} $ be a normal serie of $ S_{4} $, that $ B_{4} $ is Klein’s four-group. Since $ B_{4} $ isn't ...
1
vote
0answers
15 views

Difference between 1 (usual) and 1 bar of cayley table?

Why we write 1 as 1 bar in cayley table, instead of usual 1.
0
votes
1answer
13 views

$f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$?

Let $f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then is it true that $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$ ? Since both $f,g$ are non-trivial , I ...
1
vote
1answer
31 views

Chosen maximal subject is a subgroup

Let $ G $ is a finite soluble group and $ N $ be a unique minimal normal subgroup of $ G $. Let $ G = TS $ that $ S $ is the fitting subgroup of $ G $ and $ T = N_{G}(H) $ for $ H \leq G $. Suppose $ ...
-5
votes
0answers
32 views

The order of the derived group [on hold]

Let $G$ be a non-Abelian group. Prove that if $G/Z(G)$ is isomorphic to the dihedral group $D_8$, then $|G′|=4$.
2
votes
1answer
39 views

Subgroups of generalized dihedral groups

A generalized dihedral group, $D(H) := H \rtimes C_2$, is the semi-direct product of an abelian group $H$ with a cyclic group of order $2$, where $C_2$ acts on $H$ by inverting elements. I know that ...
1
vote
0answers
21 views

Extension of group

Let $ p = 2 $, and Let $ H $ be a subgroup of a direct product of copies of $ S_{3} $. Why $ H $ is an extension of a $ 3 $-group by a $ 2 $-group, and $ H/O_{p^{\prime}}(H) $ is a $ 2 $-group ?
1
vote
0answers
15 views

Number of sets containing m decomposable permutations of n objects.

Let $P_{m,n} = \{ \sigma_i \in S_n \}$ be a set containing $m$ arbitrary permutations of $n$ objects. Let $Q_{m,n} = \{\sigma_{ij} = \sigma_i^{-1}\sigma_j \mid \sigma_i, \sigma_j \in P_{m,n} \}$ be ...
0
votes
0answers
15 views

Reversing the search for a convex hull

In the wikipedia article on convex hulls, there is an image showing a rubber band shrinking down to form a polygon around a set of points in a plane. I have a set of data with a region with no points ...
2
votes
1answer
19 views

$ G/N $ is is a subgroup of a direct product of copies of the cyclic group $ C_{p-1} $ if $ p>2 $

Let $ G $ is soluble group and $ A $ be a unique minimal normal subgroup of $ G $. Then $ A $ is a elementary abelian group of a prime power. Let every chief factor of $ G/A $ has order $ 4 $ or a ...
1
vote
4answers
65 views

$\mathbb{Z}_6/\mathbb{Z}_2$ isomorphic to $\mathbb{Z}_3$?

Recently in class my teacher mentioned that the quotient group $\mathbb{Z}_6/\mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3$. May I ask why is this so? Also, what do elements in ...
2
votes
1answer
56 views

Possible order of $ab$ when order of $a$ and $b$ are known.

Let $a,b\in G$ be elements of a finite group $G$. We know $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. In dependence of $m$ and $n$ what are the possible values of ...
-1
votes
0answers
21 views

proofing $Z(G)=\langle [x,u]\rangle$ if $M=C_G(u)$ is maximal subgroup

Let $G$ be non-abelian finite p-group, $p$ is odd, with cyclic center and $u\in G$ be of order $p$ if $M=C_G(u)$ (centralizer of $u$) be a maximal subgroup and $Z(G)\le M$ for $x\in G\setminus M$ how ...
2
votes
0answers
24 views

calculation $p$-Fitting subgroup

Let $ G $ be a finite soluble group and $ A $ is the unique minimal normal subgroup of $ G $ that $ \vert A \vert = p^{a} $, $ p $ is prime. Let $ N =Fit(G) $, then $ N = O_{p}(G) $. Suppose $ F/N = ...
1
vote
1answer
38 views

Does such homomorphism exist?

$G$ is a group: $|G|=20$. Is there such a group G, for which the homomorphism $\tau :G-->Z_{10}$ exist?$$$$ The same question for: $\tau :G-->Z_{15}$ $$$$ I think that I should use here the ...
3
votes
2answers
41 views

Capable groups of order $32$ with GAP

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. How can I find all capable groups $G$ of order 32 with $|Cent(G)|=10$, where $Cent(G)$ is the set of all ...
0
votes
1answer
34 views

Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
3
votes
0answers
22 views

Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ ...
1
vote
1answer
39 views

How to prove that O(Ng) | O(g)

I have this exercise: Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all $Ng\in G/N$, $$o(Ng)\mid o(g).$$ For now, without using the canonic homomorphism $\tau ...
1
vote
2answers
40 views

Homomorphisms from $\mathbb{Z}_p$ to $\mathbb{Z}_3$

For which odd values of $p$ can we find a non-trivial homomorphism from $\mathbb{Z}_p$ to $\mathbb{Z}_3$ ? Is there any method to find those homeomorphisms explicitly? I have no any idea to handle ...
0
votes
0answers
35 views

Condition under which $HK$ is a subgroup

Suppose $G$ is a finite group and $H$, $K$ are subgroups. $H < N_G(K)$ is a sufficient condition for $HK$ to be a subgroup, but is is possible that $HK$ is a subgroup although neither $H$ nor $K$ ...
-1
votes
0answers
39 views

The number of all groups with n elements? [on hold]

Let $n$ be a positive integer number. How many groups of $n$ elements (which are not isomorphic) ?
-2
votes
0answers
26 views

Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $.

Let $ G $ is a soluble group and $ N $ minimal normal subgroup of $ G $. Let $ F = Fit(G) $. Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $. I show ...
1
vote
1answer
14 views

prove unique minimal normal subgroup of soluble group by $ P_{G}(M) > M $

Let $ G $ be a soluble group. If $ P_{G}(M) = \langle y\in G | \langle y \rangle M = M\langle y \rangle \rangle > M $for any subgroup $ M $ of prime power index in $ G $, then every chief factor ...
0
votes
0answers
25 views

Decomposition of certain representation of cyclic group by irreducible.

Let $C_{n}$ denotes cyclic group of order $n$. Let the set of real irreducible representations of $C_{n}$ can be listed as $$ \begin{cases} \{1,\xi,\xi^2 , \cdots \xi^{(n-1)/2}\}, \text{when $n$ is ...
3
votes
1answer
37 views

$ S_{4} $ has a sylow tower?

A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, ...
3
votes
1answer
37 views

Determining the value of A given $Z_4=Z_8\oplus Z_2/A$

Let Z define the integers and $Z_a$ define the integer group modulo a. I want to determine what A is. Given $Z_4\cong Z_8\oplus Z_2/A$, where $A\subset Z_8\oplus Z_2$, am I able to just say that ...
0
votes
1answer
21 views

Determining a lower bound on the order of a group based on its presentation

I am reading Abstract Algebra book by Dummit and Foote (3-rd edition). On pages 26-27 they define a dihedral group: $D_{2n} = \langle r,s | r^n = s^2 = 1, rs = sr^{-1} \rangle$ The authors ...
5
votes
1answer
63 views

If $|H|=112$ then $A_7\cap H \lhd H$?

I posted this because Alex Clark asked in chat and I'm not sure how to proceed. Let $G$ be a group such that it has a fixed subgroup isomorphic to $A_7$, which we denote simply by $A_7$. Let $H$ be a ...
-4
votes
1answer
58 views

I need this book by Michael Weinstein, Between nilpotent and solvable [closed]

I need this book by Michael Weinstein, Between nilpotent and solvable. I live in iran and i can't find this book. Please help me.
2
votes
0answers
42 views

$ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $.

Let $ G $ be a soluble group, $ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ for prime $ p $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $. For proof i employ the ...
0
votes
2answers
35 views

Need to prove that the A4 group is Normal sub-group of S4

I already proved that N, wich is a sub-group of S4 (4-permutations), which is all the permutations, which look's like: $(a,b)(c,d)$ (which are defintly are in A4 (even permutations of S4)) are a ...
0
votes
1answer
15 views

$ G $ is $ p $-supersolvable group . $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Show $ Q \unlhd G $.

Let $ G $ is a finite $ p $-supersolvable group for odd prime numbers . Suppose $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Now i'll show $ Q \unlhd G $. Since $ G $ is $ p $-supersolvable, then $ ...
4
votes
1answer
36 views

Showing a subset of $S_n$ is a subgroup

Let $P$ be the set of all the elements of $S_n$ which can be written as $\sigma\mu\sigma^{-1}\mu^{-1}$ for $\sigma, \mu \in S_n$. Show this is a subgroup. This doesnt seem to be as simple as ...
-1
votes
1answer
28 views

Order of Automorphis group [closed]

Let $ G $ is a finite solvable group and $ N $ be a normal minimal subgroup of $ G $ that $ G = MN $ for maximal subgroup $ M $ of $ G $, which $ M \cap N = 1 $. Let $ \vert N \vert = 4 $. Then $ ...
1
vote
1answer
30 views

Endomorphism structure of the Klein four-group

I am reading the Algebra by Grillet, this is ex 17(-18), pag. 22. I understand that $V$ can be viewed as a two dimensional vector space over $\mathbb{F}_2$. Noticed this, it is easy to see that ...
-1
votes
1answer
30 views

Direct product of quotient groups

Let $ G $ is a finite solvable group, Suppose $ H $ and $ N $ are minimal normal subgroups of $ G $. Then $ G/N \times G/H \cong G/N\cap H $ ?
0
votes
1answer
21 views

$ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $, $ G/N_{1}\cap N_{2} \cong G $?

Let $ G $ is a finite group and $ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $ that $ N_{1} \neq N_{2} $. Suppose $ G/N_{1} $ and $ G/N_{2} $ are supersolvable. Then $ G $ is ...
0
votes
2answers
41 views

How can I prove that $g\cdot H \cdot g^{-1}$ is also finite and has the same number of elements that $H$?

Suppose $G$ a group and $H$ is a finite subgroup of $G$ also $\forall g \in G$ the set $g\cdot H \cdot g^{-1}=\{ g\cdot h \cdot g^{-1} : h\in H\}$, is a subgroup of $G$. Prove that $g\cdot H ...
2
votes
1answer
73 views

How to prove that G is a cyclic Group? [duplicate]

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
-1
votes
0answers
23 views

$ G $ is soluble, then every properties of $ G $ is inherited by $ G/N $ ? [closed]

Let $ G $ be a finite group and $ G $ is soluble. Suppose $ N $ be a normal minimal subgroup of $ G $. Then every properties of $ G $ is inherited by $ G/N $ ?
7
votes
0answers
40 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
4
votes
2answers
27 views

$G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; $G$ is not simple

Let $G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; then how to show that $G$ is not simple ? I have proceeded by Cayley's theorem , $\ker f$ is ...
2
votes
0answers
16 views

Frattini subgroups and nilpotent groups: bijection? [duplicate]

I have been proved that the Frattini subgroup of a finite group is nilpotent. Now I am wondering: is the converse true? I mean, if $G$ is a finite nilpotent group, is there always a finite group ...