The study of symmetry: groups, subgroups, homomorphisms, group actions.

learn more… | top users | synonyms (2)

1
vote
3answers
93 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 ...
0
votes
0answers
25 views

Is group cohomology functorial in the first argument?

Ie, suppose you have a group $G$ acting on a group $A$ (allowing both to be nonabelian), then suppose you have another group $G'$ acting on $A$, and a group homomorphism $G\rightarrow G'$ (maybe we ...
2
votes
1answer
32 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
106 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 ...
1
vote
1answer
23 views

Normal closure of the nonnormal factor of Holomorph of a Cyclic group

Let $C_n$ be the cyclic group of order $n$. Then, we can consider the holomorph $G=C_n\rtimes Aut(C_n)$. let $H$ be such that $Aut(C_n)\leq H\trianglelefteq G$. Is it necessarily the case that $H$ is ...
1
vote
1answer
76 views

Centre of $GL(n,\mathbb{R})$ [duplicate]

Can you help me regarding the Centre of $GL(n,\mathbb{R})$ $?$ It is easy to see that the diagonals are there. what could be the other elements? It may be an useless question but it came to my mind! ...
1
vote
1answer
54 views

Residually Finite group $\Rightarrow$ Totally disconnected

How can I prove that a residually finite group $G$ is totally disconnected? I considered the topology generatad by $\{Ng\}_{N\in\eta,\;g\in G}$ where $\eta=\{N\unlhd G \;, |G:N|<\infty\}$ and I ...
1
vote
1answer
53 views

Simple question on topological groups

Why is $\{1\}$ closed in a totally disconnected topological group?
0
votes
0answers
24 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
184 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
26 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 ...
3
votes
2answers
35 views

which numbers occur as the order of elements of $(\mathbb Z /35 \mathbb Z)^*$?

which numbers occur as the order of elements of $(\mathbb Z /35 \mathbb Z)^*$? This is what I did: First I calculated that $\#(\mathbb Z / 35 \mathbb Z)^* = \phi (35) = 24 = 2^33$. Now for any ...
1
vote
0answers
31 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
22 views

$H\le G$, $G$ topological group $H$ open $\Rightarrow H$ closed

By definition if $G$ is a topological group then there exits $\nu:G\times G\rightarrow G$ defined by $(g,h)\mapsto gh^{-1}$ continous. Then Let $H\le G$ be open. Clearly $H=H\cdot 1^{-1}=1\cdot H$. ...
3
votes
3answers
134 views

$G \times H \cong G \times K$ , then $ K \cong H$

I already know that if groups $G,H,K$ are finitely generated abelian groups, following is true. If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. I prove this by uniquness ...
3
votes
1answer
53 views

The group acts on an ordered set

Let $G$ be a group, $G$ acts on an ordered set and preserves its order, i.e. $a<b$, then $g(a)<g(b)$ for $g\in G$. Then does it imply there is a left order on $G$, i.e. $f<g$, then $fh<gh$ ...
4
votes
0answers
119 views

Question on Infinite Abelian Group [closed]

If $G$ is an infinite abelian group such that all subgroups have smaller order, then I want to show that $G$ is isomorphic to $\mathbb Z_{p^\infty}$. Any suggestion?
0
votes
1answer
17 views

Group of “small”/“local” transformations

This is largely a question about terminology. I will introduce it with a simple example. Consider a point $x$ on the real line and a function $f:\mathbb{R}\to\mathbb{R}$. Normally we say that $f$ is ...
2
votes
1answer
61 views

On group whose inner automorphisms groups of order 6

Let $G$ be a finite group such that whose center is of odd order and the order of inner automorphisms groups is 6. Then prove that $G$ has a non abelian normal subgroup of order 6. Thank you
0
votes
0answers
47 views

How does the first group cohomology classify torsors?

I know this is true, but I'm having some trouble finding any references on this. I'm in particular interested in the nonabelian case. Specifically, let $G$ be a group acting on a group $A$ (both ...
0
votes
1answer
44 views

If $C(G)$ is the commutator of $G$, then $C(G\times H)=C(G)\times C(H)$?

This is a yes or no question. I have proved this myself and i'm not sure whether i'm wrong. Let $C(G)$ denote the commutator subgroup of a given group $G$. Let $G,H$ be groups. Then ...
0
votes
1answer
49 views

How do i show that finite abelian group is solvable?

Let $G$ be a finite abelian group. How do i show that $G$ is solvable using Fundamental theorem of finite abelian groups?
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 ..
0
votes
1answer
14 views

Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
2
votes
2answers
29 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
25 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: ...
1
vote
3answers
94 views

How do i prove this this group is isomorphic to this?

How do I prove that $\mathbb{Z}\times\mathbb{Z} / \langle(1,2)\rangle \cong \mathbb{Z}\times \Bbb Z_2$? I was trying to find a corresponding homomorphism $\phi:\mathbb{Z}\times\mathbb{Z}\rightarrow ...
2
votes
1answer
38 views

Group Theory Homomorphism

is there group homomorphism surjective between any group and it's subgroup ? i.e. if G is a group and H is a subgroup of G , is there f:G -> H such that f is group homomorphism surjective ?
3
votes
2answers
35 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
1answer
13 views

Show that there are 1120 elements of $S_8$ having disjoint cyclic decomposition of the type (a b c)(d e f)(g h)

Show that there are 1120 elements of $S_8$ having disjoint cyclic decomposition of the type (a b c)(d e f)(g h). Find also the number of elements in the orbit of (a b c)(d e f)(g h). My reasoning to ...
6
votes
3answers
311 views

Show that a group of order $p^2q^2$ is solvable

I am trying to prove that a group of order $p^2q^2$ where $p$ and $q$ are primes is solvable, without using Burnside's theorem. Here's what I have for the moment: If $p = q$, then $G$ is a $p$-group ...
2
votes
3answers
56 views

Are there finite-dimentional unital associative algebras over $\Bbb{C}$ that are not isomorphic to a group algebra $\Bbb{C}[G]$ for finite group G?

I've seen somewhere in my lecture notes that the answer is yes, but I can't think of an explicit example of such an algebra. Is there a standard way of constructing these algebras for particular ...
2
votes
1answer
179 views

Is the dihedral group $D_n$ nilpotent? solvable?

Is the dihedral group $D_n$ nilpotent? solvable? I'm trying to solve this problem but I've been trying to apply a couple of theorems but have been unsuccessful so far. Can anyone help me?
3
votes
1answer
71 views

The abelian group of homeomorphism

Let $G$ be a subgroup of the group of homeomorphisms on the circle, and we suppose $G$ is abelian, if every element of $G$ has a fixed point on the circle, does it imply that $G$ has a common fixed ...
0
votes
1answer
38 views

Show that $ord((2 mod 61)^n)=60$ $\iff$ $gcd(60,n)=1$

I have an old exam with this question: Given $(\mathbb{Z}/61\mathbb{Z})^{*}$ show that for $n \in \mathbb{N}$ we have: $ord((2 \mod 61)^n)=60$ $\iff$ $gcd(60,n)=1$ Update: I have something: ...
1
vote
2answers
36 views

Let $G$ be a group of order $p^n, p$ prime. Then $G$ has a composition series such that all its composition factors are of order $p$.

This is an example from a book. But I don't understand the solution. Let $G$ be a group of order $p^n, p$ prime. Then $G$ has a composition series such that all its composition factors are of order ...
0
votes
1answer
33 views

the group acts faithfully on the line

Let $G$ be a group. $G$ acts faithfully on the line $\mathbb{R}$ by orientation preserving homeomorphism, then does it imply $G$ is left ordered, i.e. there is an order $<$ on $G$, and if $a<b$, ...
2
votes
2answers
36 views

Any Sylow $p$-subgroup of $GL_2(p)$ is a Sylow $p$-subgroup of $SL_2(p)$. Is this true?

I know that a Sylow $p$-subgroup of $GL_2(p)$ has order $p$ and so does a Sylow $p$-subgroup of $SL_2(p)$. Does this mean that $Syl_p(GL_2(p)) = Syl_p(SL_2(p))$?
1
vote
1answer
31 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 ...
4
votes
3answers
106 views

Group theoretical characterization of $\mathbb Q$ and $\mathbb Q^\star$

$\mathbb Z$ can be characterized in group-theoretic language as the infinite cyclic group. But what kind of a group is $\mathbb Q$ under addition? What about $\mathbb Q^\star$? What kind of subgroup ...
3
votes
1answer
52 views

Order of finite groups with automorphism groups of prime order

Let $G$ be a finite group such that $|Aut(G)| = p$, where $p$ is a prime number. Prove that $|G| \le 3$ I found out that in this case, $G$ must be Abelian. However, I'm having trouble showing ...
5
votes
2answers
45 views

Quotient group $Z^n/\textbf{im}(A.)$

Let $A$ be matrix $n \times n$ with integer coefficients and nonzero determinant. Can we say something about $ \mathbb{Z}^n / \ \textbf{im}( \phi )$ ($\phi : v \mapsto Av$ )? This problem has arised ...
1
vote
1answer
33 views

Quandle properties

Let $G$ be a group, take Conjugacy class of $g$, let us denote it by $g^G$ and let $g^G=X$ Let $x, y \in X$, define $xy=xyx^{-1}$, then $X$ is closed under this operation How I can proof this axiom ...
0
votes
1answer
36 views

What can be assumed for $\forall m \in K\cap H$?

If we are given groups $K$, $H$ and $G$, such that $H\trianglelefteq G $ and $K \leq G$. Prove that $K\cap H \trianglelefteq G$. My work: Let $g \in G$ and $m \in K\cap H$. Since $m \in H$ and $m ...
0
votes
0answers
37 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 ...
0
votes
3answers
24 views

Proving $g( (x^{k})^{n})g^{-1}\in K$

If we are given groups $K$, $H$ and $G$, such that $K \leq H\trianglelefteq G $ and $H$ is cyclic. Prove that $K\trianglelefteq G $. My work: Let $H=\langle x \rangle, x \in G$ and $K=\langle x^{k} ...
0
votes
2answers
48 views

Is it trivial that $C_G(H)$ is normal when $H$ Is normal?

Let $G$ be a group. It is written in my text that there is a homomorphism $\phi:G\rightarrow Aut(H)$ where $H$ is a normal subgroup of $G$ and $\ker(\phi)=C_G(H)$. From this, i realize that $C_G(H)$ ...
0
votes
1answer
45 views

How to extend a member of $Aut(G/N)$ to a member of $Aut(G)$

Suppose that $G$ is a finite group with a normal subgroup $N$. Assume in addition that $\beta$ is an automorphism of group $G/N$. My question is : Under which conditions we can assure that there is a ...
2
votes
1answer
55 views

If a group doesn't have subgroups of index 2 and 3, then any subgroup of index 4 is normal.

Let $G$ be this group and $H$ be any subgroup of index 4. $G$ acts on the set of left cosets of $H$ in $G$, which is a homomorphism $\varphi: G\to Aut(G/H) = S_4$. It is easy to see that $\ker ...
0
votes
1answer
56 views

The center of Sylow $p$-subgroups of a finite simple group of Lie type

Would some one please to introduce me an easy reference!! which contains the size of $Z(P)$(center of $P$), where $P$ is a Sylow $p$-subgroup of a finite group of Lie type over a finite field of ...