0
votes
0answers
6 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a.a^p$ Let $a.a^p = b.b^p$ ...
0
votes
0answers
9 views

regarding the groups that how they form and what may be the practicaliy operations those taken to define the groups [on hold]

I studied the group theory and also the ring theory that some algebraic structures those satisfying some algebraic operations are known so. But some cases I found that some structures form a group ...
1
vote
2answers
37 views

$N$ a normal subgroup of $G$ with $|G|$ odd and $|N|=5$ satisfies $N \subset Z(G)$

Exercise Let $N$ be a normal subgroup of $G$. Suppose that $|N|=5$ and that $|G|$ is odd. Show that $N \subset Z(G)$. I am sorry for not writing any work of mine but I really don't know where to ...
0
votes
0answers
22 views

Generalization of Lattice Theorem

I've shown that Lattice Theorem is still true if I substitute the projection to the quotient with a generic surjective homomorphism, i.e. if $G$ and $G'$ are two groups and $f:G\to G'$ is a surjective ...
2
votes
1answer
36 views

A group $G$ with $|G|=p^3q$ and with no normal Sylow subgroups is $G \cong \mathbb S_4$

Problem Show that if $|G|=p^3q$ and $G$ has no normal Sylow subgroups, then $G \cong \mathbb S_4$ The attempt at a solution By the Sylow theorems we have: -$n_p \equiv 1 (p), \space n_p|q$ -$n_q ...
2
votes
1answer
25 views

Calculating number of conjugacy classes for a prime power group $G$

$\space$Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$. Since $Z(G) \neq G$, then $G$ is not abelian. I'll ...
2
votes
2answers
28 views

Is an elementry abelian group a non-degenerate symplectic vector space?

Let $A$ be an elementry abelian group with $|A|=p^{n}$ where $p$ is a prime number and $n$ is even. It is well-known that we can consider $A$ as a vector space of dimension $n$ over the field $F_p ...
1
vote
1answer
38 views

$x \rightarrow x^n$ is a group automorphism of a finite abelian group G [on hold]

How do we prove that the map $\phi:G \rightarrow G$ defined by $\phi(x) = x^n$ for some $n \geq 0$ is a group automorphism of $G$ if $\gcd(|G|,n)=1$?
2
votes
1answer
36 views

How to prove that a subgroup of a group is normal based on generating sets?

I apologize if this is a duplicate question, but I read online that one method by which to show that a subgroup is normal is by means of generating sets (if both groups have known presentations). In ...
1
vote
2answers
28 views

Order of a permutation

What does the order of a permutation actually mean? I accept the fact that it is the l.c.m. of the lengths of the cycles in its cycle decomposition, but I don't really have an intuition for what the ...
0
votes
0answers
44 views

Realize groups as unit group of a ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
1
vote
2answers
20 views

Constructing group homomorphisms.

Let $G$ and $H$ be groups, and suppose I want to construct a group homomorphism $\phi$ between them. From what I know, I just need to send each element $x \in G$ to an element $y \in H$ such that the ...
1
vote
0answers
38 views

Property of group G with $|G|=2n$ with $n$ elements of order $2$ (Sylow theorem application)

Suppose $G$ is a group such that $|G|=2n$, $G$ has $n$ elements of order $2$ and the rest of the elements form a subgroup $H$. Show that $H \lhd G$ and $n$ is odd. I am pretty lost with this ...
1
vote
0answers
33 views

How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
0
votes
0answers
32 views

Self normalising p sylow

When are p-sylow subgroups self normalising? I know, for example, that if the group has order $ p^2q^2$ then the p-sylow subgroups are self-normalising if there are $q^2$ of them. I just don't know ...
1
vote
1answer
43 views

Determining if a set is a group

Let $S=\lbrace x+y\sqrt2 : x,y\in \mathbb R \rbrace$ \ $\lbrace0\rbrace$. Justify whether $S$, together with traditional multiplication, is a group. I've verified that the set is closed under the ...
0
votes
0answers
35 views

Isomorphism of a set [duplicate]

We know that $\operatorname {Aut}(G) \over \operatorname {Inn}(G)$ $\cong \operatorname {Out}(G)$. Is it true that $\operatorname {Aut}(G) \cong \operatorname {Inn}(G) \rtimes \operatorname {Out}(G)? ...
1
vote
1answer
45 views

automorphism group of groups [on hold]

Given a group $G$, I would like to calculate $\operatorname{Aut}(G)$. From definition of $\operatorname{Aut}()$ we know: $\operatorname{Aut}(G)\le \operatorname{Sym}(G) $ If the group is finitely ...
2
votes
1answer
42 views

Transitive action on two sets! [duplicate]

Suppose $G$ is a finite group and G acts transitively on sets $X$ and $Y$. Let $a$ and $b$ belongs to $X$ and $Y$ respectively and $G_{a}$ be stabilizer of $a$ in $X$ and $G_{b}$ be stabilizers of ...
2
votes
1answer
44 views

Why the paired orbit has the same size here?

enter link description here On this proof, he just showed that the the paired orbit of (a,b) has the same size, but this seem has nothing to do with the size of corresponding paired orbit of an orbit ...
1
vote
1answer
69 views

Some functorial maps $G\times G\rightarrow G$

Let $G$ be a group. Le diagonal map $\delta:G\rightarrow G\times G$ obviously gives a functorial morphism from the identity functor of $\mathbf{Grp}$ to the functor $P$ sending $G\mapsto G\times G$ ...
1
vote
2answers
42 views

Group of order $pq$ with $p\not\mid (q-1)$

Let $p, q$ be prime numbers, with $p<q$. If $G$ is a group of order $pq$ and $p\not\mid (q-1)$, then $G\cong \mathbb{Z}/pq\mathbb{Z}$. The standard way to prove this fact is using Sylow theorems, ...
0
votes
0answers
23 views

Crossed homomorphism: $\varphi(x)=\varphi(y)\iff Kx=Ky$

Let $G$ be a finite group, $N\unlhd G$. A crossed homomorphism from $G$ to $N$ is defined as a $\varphi:G\to N$ s.t. $\varphi(xy)=\varphi(x)^y\varphi(y)$. It's not in general a group homomorphism. ...
-1
votes
0answers
68 views

Is $\text{Aut}(A \times B) = \text{Aut}(A) \times \text{Aut}(B)$? [on hold]

Is $\text{Aut}(A \times B) = \text{Aut}(A) \times \text{Aut}(B))$? where $A$ and $B$ are subgroups, $A \times B$ is the direct product and $\text{Aut}(A)$ refers to group of all automorphisms of ...
1
vote
1answer
18 views

If $\sigma \in Aut(G)$where $|G|$ odd has order 2 there exists Sylow $p$-subgroup with $\sigma(P)=P$

Let $G$ be a group of odd order and $\sigma$ an automorphism of G of order 2. Show that if the prime $p$ divides $|G|$ then there exist a Sylow $p$-subgroup $P$ such that $\sigma(P)=P$.
1
vote
1answer
62 views

Automorphisms of the group of integers $\mathbb Z$

Can anyone help me showing $\operatorname{Aut}(\mathbb Z)\simeq \mathbb Z_2$? I guess I should define an homomorfism $\phi:\mathbb Z\longrightarrow S(\mathbb Z)$ with kernel $2\mathbb Z$ and image ...
2
votes
1answer
41 views

Order of $a^m$.

Let $G = \langle a \rangle$ a finite cyclic group of order $n$. Prove that $|a^m| = \frac{n}{\gcd(m,n)} = \frac{\mathrm{lcm} (m,n)}{m}$. I managed "half" of it. Write $|a^m| = k$ and $d = ...
5
votes
1answer
75 views

How many automorphisms does $S_3\times S_3$ have?

I've shown that $|\text{Aut}(S_3\times S_3)|\ge 72$, how can I show that $|\text{Aut}(S_3\times S_3)|\le 72$ ?
1
vote
1answer
37 views

Let $H\le G$ s. t. whenever $Ha≠Hb$ then $aH≠bH$. Prove that $gHg^{−1}\le H\;$ $\forall g\in G$. [on hold]

Suppose that H is a subgroup of G such that whenever$ Ha \ne Hb $ then $ aH \ne bH $. Prove that $ gHg^{-1} \subseteq H$ for all g in G.
2
votes
1answer
33 views

Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
1
vote
2answers
32 views

Orbit and stabaliser of $2\times2 * 2\times1$ matrices

I have the group action of matrix multiplication, meaning: $g((x,y))=\begin{pmatrix}a&0\\0&b\end{pmatrix}$$ \begin{pmatrix}x\\ y\end{pmatrix}$ ...
2
votes
0answers
54 views

If $X$ generates $\Bbb{Q}$ then $X\setminus\{x\}$ also generates $\Bbb{Q}$ [duplicate]

If $X$ is a generator subset of $\Bbb{Q}$ then for $x\in X$, $X\setminus\{x\}$ also generates $\Bbb{Q}$. Clearly if I can express $x$ as a combination of the remaining generators we are done. ...
0
votes
1answer
50 views

Normal subgroup of a finite group

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?
1
vote
1answer
35 views

what are the p-orbits in the decomposition of Σ into p-orbits.

At the bottom in the proof of slow 2, what's the meaning of "restrict the action of G on Σ to an...on Σ"? Since I can't understand that so I don't know what are the p-orbits in the decomposition of Σ ...
1
vote
2answers
67 views

Is orbit a group?

Let a group $G$ act on a set $S$, and let $s$ be an element in $S$. The identity of the orbit of $s$ is $s$ itself and if $a$, $b$ are in the orbit of $s$, then $a b = g_1 g_2 s$, where $g_1$, $g_2$ ...
1
vote
1answer
37 views

Rapid question on minimal normal subgroups

Why does $N$ finite group, $M\unlhd N$ imply that $M$ contains a minimal normal subgroup of $N$? If $M$ is itself minimal in $N$ we have finished. But if not, we know that there exists a minimal ...
4
votes
1answer
29 views

Product of two stabilizers of transitive group action is proper subset of G?

Suppose $G$ is a finite group and G acts transitively on some set $X$. Let $a$ and $b$ be two distinct elements of $X$ and $G_{a}$ and $G_{b}$ be stabilizers of $a$ and $b$ respectively.Show that ...
-1
votes
0answers
22 views

What is the group $_nG$?

I've seen the notation $_nG$, where $G$ is a group and $n > 1$ an integer. What does this notation mean? I suspect it has to do with torsion somehow. Maybe a notation for the $n$-torsion subgroup ...
0
votes
0answers
31 views

solvable group and derived series [duplicate]

Can someone prove that group is solvable if and only if its derived series terminate on a trivial group? http://en.wikipedia.org/wiki/Solvable_group
1
vote
3answers
38 views

Proving that a set is a group under addition

To show that a set $G$ is a group under addition, do we first need to show that $G$ is closed under addition, or is that implied by proving the three properties of a group, namely there exists an ...
2
votes
1answer
55 views

Can you explain more specifically about there is no simple group of size 96

For the solution,I know that the number of Sylow-$2$ subgroups, $n_2=3$, and we can find a subgroup of order $32$,then $G$ can act on the left cosets of $H=P_2$, then it gives a map from $G$ to $S_3$, ...
1
vote
1answer
49 views

Quotient group $G/\{1\}=G$ if $1$ is the identity element of $G$

Is it true that quotient group $G/\{1\}=G$? Or isomorphic to $G$?
3
votes
0answers
43 views

Why are centers, centralizers and normalizers called that way?

I know what they are, but where do the names come from?
0
votes
1answer
36 views

Upper bound for the number of generators of a group

Let $H\leq G$ and $x\in G$. If $H$ is generated by at most $n$ elements, prove that $\langle H,x\rangle$ is generated by at most $n+1$ elements. This is intuitively obvious but when I try to ...
3
votes
1answer
49 views

Direct product of two groups.

Let $N$ be a minimal normal subgroup of $G$. Also let $N$ and $\frac{G}{N}$ are non-abelian simple. Can we say that $G=N\times A$ where $A$ is a non-abelian simple subgroup of $G$?
1
vote
1answer
42 views

non-abelian simple group

Let G be a non-solvable group and $(\frac{G}{Z(G)})^{'}=\frac{G}{Z(G)}$. Can we say that there is a normal subgroup $N$ of $G$ with property $Z(G)\leq N$ such that $\frac{G}{N}$ is a non-abelian ...
1
vote
3answers
83 views

Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?

Is the following statement true or false: If $G$ is a group with the property that $g=g^{-1}$ for all $g \in G$, then $G$ is abelian. I believe it is false since I know that abelian ...
-2
votes
3answers
58 views

If $N$ normal subgroup of $G$ and $M$ normal subgroup of $G$ prove $MN$ is a subgroup [closed]

If $N\vartriangleleft G$ and $M\vartriangleleft G$ and $MN = \{mn | m \in M, n \in N\}$, prove that $MN$ is a subgroup of $G$ and that $MN\vartriangleleft G$
0
votes
1answer
22 views

Abstract Monomorphism 3 part Question

I have been working on this problem for an hour now and gotten nowhere: Let $G$ be any group and $A(G)$ the set of all 1-1 mappings of $G$, as a set, onto itself. Define $L_a : G \rightarrow G$ by ...
1
vote
1answer
41 views

If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$

If $G$ is a finite group and its $p$-sylow subgroup $P$ lies in the center of $G$, prove that there exists a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$