-2
votes
0answers
44 views

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? [on hold]

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? Does there exist a subgroup of $G$ of order $180$?
2
votes
1answer
31 views

Lower central series and the center of $S/S^k$

Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. I don't know much about lower central ...
2
votes
0answers
61 views

Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
0
votes
0answers
14 views

Prove that both $R_p$ and $R^p$ are abelian groups under ordinary addition of rationals.

Q: Let $p$ be a fixed prime. Let $R_p$ be the set of all those rational numbers whose denominator is relatively prime to $p$. Let $R^p$ be the set of rationals whose denominator is a power of $p\ ...
2
votes
2answers
147 views

Show that $\mathbb{Z}_p\setminus\{\overline{0}\}$ is not a group if $p$ is not prime.

The answer is too short that I think I've gone wrong at some point! Q: If $p$ is prime, then the nonzero elements of $\mathbb{Z}_p$ form a group of order $p-1$ under multiplication. Show that this ...
1
vote
1answer
47 views

I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
0
votes
1answer
53 views

Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
0
votes
1answer
45 views

Group theory problem

I am asked to prove "Show that if $${e}<H_1<H_2<...<H_{n-1}<G$$ Is a subnormal series for a group G, and if the order of $H_{i+1}/H_i=s_{i+1}$, then G is of order $s_1 ...
2
votes
1answer
32 views

For a group $G$ acting transitively, is $G_{\alpha}$ contained in stabilizer of some block $\Delta$ if $\alpha \in \Delta$?

This question refers to permutation groups, in particular the primitive ones, and block systems. Let $G$ be a finite group acting on a set $\Omega$, and consider some partition $\Delta_1 \cup \ldots ...
4
votes
1answer
57 views
+100

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
2
votes
1answer
30 views

The minimum size of generating set of the external direct product

I have seen the following theorem here: Suppose that $A$ and $B$ are finite groups whose orders are relatively prime to each other, and the minimum size of generating set (i.e., the smallest ...
7
votes
1answer
133 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq ...
2
votes
1answer
55 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
4
votes
1answer
116 views

Do these definitions make sense?

Letting $G$ be a group and $S(G)$ be all permutations of $G$, define $$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$ It is easy to check that $L(G)$ is a ...
1
vote
1answer
51 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
4
votes
2answers
50 views

How is this subgroup abelian?

Let $G$ be a finite group of order $2n$ such that half of the elements of $G$ are of order $2$ and the other half form a subgroup $H$ of order $n$. Then I know that $H$ is of odd order because for ...
1
vote
1answer
56 views

Generating a subgroup

While getting ready for my exams I am trying to solve this question "Find a Sylow $2$-subgroup in $S_4$ which contains the element $(1, 3, 2, 4)$." I started by permutation of the element required ...
-1
votes
0answers
30 views

Group Structure on the set of matrices over finite field

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. You must be agreed with me that the set $M_n$ ...
0
votes
1answer
47 views

Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
1
vote
1answer
46 views

Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
5
votes
1answer
65 views

Prove that $G$ has a nontrivial normal subgroup

I wanted to ask if I had done this problem correctly. Let $G$ be a group of order $pqr$ (for $p > q > r$ primes). (i) If $G$ fails to have a normal subgroup of order $p$, determine the ...
1
vote
1answer
41 views

Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$. The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, ...
1
vote
0answers
37 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
7
votes
1answer
118 views

Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian

Suppose that $G$ is a finite group. If $P( ab=ba ) >5/8$, prove $G$ is abelian.
1
vote
2answers
43 views

Let $G=\mathbb Z_4 \times \mathbb Z_2$. Find all $H$ subgroups of $G$ of order 2, so that $G / H$ is cyclic.

Let $G=\mathbb Z_4 \times \mathbb Z_2$. Find all $H$ subgroups of $G$ of order 2, so that $G / H$ is cyclic. I tried going brute force, and realized there are 3 candidates: $H_1=\{(0,0), (2,0) \}$ ...
0
votes
2answers
179 views

Total number of non isomorphic groups of order 122.

Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order ...
4
votes
3answers
95 views

Show that number of solutions satisfying $x^5=e$ is a multiple of 4?

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5 = e$ is a multiple of 4. I think that besides $ x, x^2, x^3, x^4 $ also satisfies the given equation but ...
3
votes
1answer
48 views

Characterizing finite non-abelian groups in which every subgroup is abelian

How to prove: A non-abelian finite group in which every subgroup is abelian has order divisible by at most two primes.
2
votes
3answers
43 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
0
votes
1answer
26 views

Are augmented algebra maps of group algebras group homomorphisms?

Given a finite group $G$ and a field $k$, we can form the group algebra $kG$ with basis the elements of $G$. There is a natural augmentation $\varepsilon\colon G\to k$ that sends an element to the sum ...
5
votes
3answers
50 views

Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
9
votes
0answers
116 views

Is there any simple proof for this?

Let $G$ be a group with $|G:Z(G)|=n$ then $\phi(x)=x^n$ is a homomorphism from $G$ to $Z(G)$. I guess it has a proof using transfer theory, I wonder whether it has an elemantary proof or not. ...
3
votes
3answers
30 views

Derived Subgroup and Factor Groups

Let $N \unlhd G$, does it holds that $(G/N)' = G'N /N$, or more generally for any subgroup $H \le G$ we have $(HN/N)' = H'N/N$? Does anyone has a proof of this fact? PS: Side-Question: Is it wrong to ...
1
vote
0answers
24 views

Poincare polynomial of a finite $G$-module with $G$ being a $p$-group

Recently, I've been reading Shatz's book, profinite groups, arithmetic, and geometry. Let $G$ be a finite $p$-group and $A$ a finite $G$-module such that $pA=(0)$. In the proof of Theorem 19 (p.82), ...
1
vote
1answer
33 views

Is $H/H_0 = HG/H_0G$?

If I have a quotient group $H/H_0$ and another group $G$ such that $H_0G\unlhd HG$. Is it then true that $H/H_0 = HG/H_0G$?
1
vote
3answers
66 views

Two 3-cycles generate $A_5$

I want to solve the following exercise, from Dummit & Foote's Abstract Algebra Let $x$ and $y$ be distinct 3-cycles in $S_5$ with $x \neq y^{-1} $. Prove that if $x$ and $y$ do not fix a ...
2
votes
1answer
30 views

The largest normal subgroup of odd order in the centraliser

For a group $G$ denote by $O_{2'}(G)$ the largest normal subgroup of $G$ of odd order. Now let $N = O^{2'}(G)$ be the smallest normal subgroup of $G$ such that $G/N$ has odd order, and $H \le G$ be ...
3
votes
4answers
102 views

An elementary problem related to $Z(G)$

Let $G$ be a group, If $xy\in Z(G)$ then $C_G(x)=C_G(y)$. Note: Even it is very elementary, I liked it. Edit: Thanks for different solutions, you may want to examine the case if $xyz\in Z(G)$ then ...
0
votes
0answers
22 views

Normalised subgroup and $O_{2'}(G)$ and $O^{2'}(G)$.

Let $G$ be a finite group. Denote by $N = O^{2'}(G)$ the smallest normal subgroup, such that $G / N$ has odd order. Let $H$ a subgroup of odd order which is normalised by every Sylow $2$-subgroup of ...
9
votes
1answer
166 views

Largest symmetric group contained in alternating group

I know that for $n \geq 3$, the alternating group $A_n$ contains a subgroup which is isomorphic to $S_{n-2}$, namely $$\langle \{(i \;i+1)(n-1 \;n):1 \leq i \leq n-3\} \rangle.$$ I was wondering what ...
0
votes
0answers
24 views

Conclusion about conjugacy classes of $\pi$-complements

Let $\pi$ be a set of primes, and let $N$ have a single conjugacy class of $\pi$-complements and these are nilpotent. If $H^G / N$ is a $\pi$-group, then $H^G$ also has a single conjugacy class of ...
1
vote
0answers
26 views

Question on Proof involving direct products and $\pi$-complements

All groups $G$ are finite, for $N \unlhd G$ we set $\overline{G} := G/N$ and for some $U\le G$ we write $\overline U = UN/N$. Also for a set $\pi$ of primes we denote by $O_{\pi}(G)$ the largets ...
1
vote
1answer
33 views

If $G'\leq Z(G)$ implies $G'=Z(G)$?

If we know that $G'$ is a subgroup of a $Z(G)$, can we conclude that $G'=Z(G)$ ? What I can see is that $G/Z(G)$ is abelain and it can not be cyclic but no further ..
2
votes
2answers
25 views

Part of Proof which implies normal complement

"Let $N := O^{2'}(G)$, then $N \unlhd G$ is a normal subgroup, and suppose that $C_G(N) = G$. Then $G$ has a central Sylow $2$-subgroup. So $G$ has a normal $2$-complement." I do not understand the ...
0
votes
1answer
24 views

Let $H \le G$, when does there exists normal complements.

Let $H \le G$, if $H$ is normal in $G$ and $(|H|, |G/H|) = 1$, then according to the famous Schur-Zassenhaus-Theorem we can find a complement in $G$ for $H$. This need not be normal. Now my question: ...
2
votes
2answers
26 views

Some basic terms from finite group theory, normalising and centralising

In a proof a read the following "Since $H$ and $N$ normalize each other, if follows that $H \subseteq C_G(N)$". I thought normalising just means that one subgroup is normal with respect to the other, ...
0
votes
0answers
28 views

$|G|=12$ , three Sylow 2-subgroups exist. Show that $|\{a \in G | ord(a)=2 \} | \geq 7$

$|G|=12$ , three Sylow 2-subgroups exist. Show that $|\{a \in G | ord(a)=2 \} | \geq 7$ Obviously, there exist three elements of order $2$. How can I find the remainder?
1
vote
1answer
23 views

Every finite group has a composition series

Every finite group has a composition series. The proof of this statement is as follows Proof. If $|G| = 1$ or $G$ is simple, then the result is trivial. Suppose $G$ is not simple, and the result ...
1
vote
3answers
35 views

How many Sylow-$ 3$ subgroup does $G$ have?

Let $G$ be a noncyclic group of order $21$. How many sylow-$3$ subgroup does G have? The possible orders of Sylow $3$ subgroups is $1, 7$. But how to check the exact number?
0
votes
0answers
34 views

Relation between complements, $p$-complements and $\pi$-complements

Let $G$ be a group, a few definitions: 1) If $H \le G$, then a complement of $H$ in $G$ is a subgroup $K \le G$ such that $$ G = HK \quad \mbox{and} \quad H \cap K = \{ 1 \}. $$ 2) Let $p$ a prime ...