Tagged Questions
1
vote
0answers
49 views
Relationship between curls, gradients, and divergences; and the Isomorphism Theorem
I am trying to develop a geometric intuition for the relationship between the curl, the gradient, and the divergence based on the Isomorphism Theorem, where the Isomorphism Theorem says that "If ...
0
votes
1answer
43 views
ordering of a group
An ordering of a group $G$ is a linear ordering $<$ on (the underlying set of) $G$
that satisfies, in addition,
$a < b \implies ac < bc ∧ ca < cb$,
for all $a,b, c \in G$.
Show that a ...
1
vote
0answers
32 views
A strange characterisation of cyclic groups [duplicate]
"A finite group is cyclic if, for any integer m, the number of elements
of order dividing m is at most m."
I have never seen this characterisation of cyclic groups before. How do I prove this? I hope ...
4
votes
2answers
61 views
Cardinality of $GL_n(K)$ when $K$ is finite
I don't know how to do the last task of an exercise.
Let $K$ be a field, $G=GL_n(K)$ and $X=K^n\backslash\{0\}$.
First task: Show that $G \times X \to X$, $(A,x)\mapsto Ax$ defines an action of $G$ ...
-1
votes
0answers
54 views
Orbits of group actions
I have the following problem:
Describe the orbits of the group action in each of the following cases (you are not asked to show they are actions):
(a) $(0,\infty)$ acts on C by multiplication
(b) ...
0
votes
1answer
61 views
1
vote
2answers
74 views
$H$ must contain every Sylow $p$-subgroup of $G$
G is finite and has a prime factory. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that
4
votes
2answers
60 views
If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.
If the order of a finite abelian group is square free, show that the group is cyclic.
This is a question from "basic abstract algebra" by bhattacharya
3
votes
3answers
72 views
Sum of two squares in a $\Bbb Z/p\Bbb Z$
I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of ...
2
votes
3answers
61 views
Permutations and Cross-ratios
Pick four distinct numbers, list all 24 permutations, and compute the cross-ratio of each permutation. Show that at most six numbers have occurred, given by the cross-ratio group
$y, \frac{1}{y}, 1-y, ...
10
votes
2answers
118 views
Group of invertible elements
Let R be a ring with unity. How can I prove that group of invertible elements of R is never of order 5? My teacher told me and my colleagues that problem is very hard to solve. I would be glad if ...
0
votes
1answer
27 views
Showing that $V_4 \supset [A_4,A_4]$
I try to show that $V_4 \supset [A_4,A_4]$.
Just by trial and error I found that $(12)(34)=[(123),(124)]$.
I think I can find the other two also by trial and error, but is there also an smarter ...
2
votes
0answers
29 views
Alexanderpolynomial of torus knot
i want to compute the Alexanderpolynomial of the torus knot $T_{p,q}$ with $p$ and $q$ coprime. I should work with the groups presentation $G(T_{p,q})=<x,y:x^p=y^q>$ of $T_{p,q}$. I have to use ...
2
votes
3answers
48 views
Showing that this set of functions is a group.
I have trouble understanding the following task:
Show that the set $X = \{f_1,\ldots ,f_6\}$ of functions $f_i : \Bbb Q\setminus \{0,1\} \to \Bbb Q\setminus \{0,1\}$ with
$x ↦ f1(x) = x$,$x ↦ ...
1
vote
2answers
70 views
Is G isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$?
If $ G=\{3^{m}6^{n}|m,n \in \mathbb{Z}\}$ under multiplication then i want prove that this G
is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$.Can any one help me to solve this example?
please help me. ...
2
votes
3answers
115 views
Which group is isomorphic to $\left\langle\begin{bmatrix}0&1\\1&0\end{bmatrix},\begin{bmatrix}1&-1\\0 & -1 \end{bmatrix} \right\rangle$?
Both matrices have determinant equal to -1, so their products are matrices with determinant $\in \{1,-1\}$. Can I conclude that this is isomorphic to $ O_2(\mathbb{R}) $ ?
3
votes
1answer
39 views
order of $\langle (123) , (234) \rangle$
As homework the teacher asks us to determine how many elements are there in $\langle (123) , (234) \rangle \subset S_4$ .
I've started doing all the multiplications between the elements, and I've ...
0
votes
0answers
32 views
$\langle v,\sqrt{2}v\rangle_{\mathbb{Z}}$ not a discrete subgroup of $\mathbb{R}^{2}$ [duplicate]
I got a list of exercises to do and there is one of the first exercises which I do not manage to solve.
Its statement is the following:
Let $v\in \mathbb{R}^{n}$ be a nonzero vector. Using the fact ...
2
votes
1answer
32 views
Unable to get to standard permutations after $n-1$ transpositions
Problem: Give an example of a permutation of the first $n$ natural numbers from which it is impossible to get to the standard permutation $1,2,\ldots,n$ after less than $n-1$ transposition operations ...
1
vote
1answer
67 views
$\bigcup_{x \in G} xHx^{-1} \neq G$
I could not solve it properly: $\bigcup_{x \in G} xHx^{-1} \neq G$ if $G$ is a finite group and $H$ is a proper subgroup (of $G$).
I tried to use the class equation and to create other actions to ...
4
votes
1answer
57 views
Action of $S_4$ in $S_4/S_3$
Let $G = S_4$, $H = S_3$, $X = G/H$ be the set of right cosets of $H$, $x = (14)H$ and $G $ acts on $X$ by conjugation. Compute $\mathscr{O} (x)$ and $G_x$ (the stabilizer of $x$).
I've got a ...
4
votes
2answers
125 views
Proper subgroup of $\mathbb{Q}^{+}$ with finite index
Is there a non-trivial subgroup $H$ of $\mathbb{Q^{+}}$, such that $|\mathbb{Q^{+}} : H|$ is finite? Of course, $|H| = \aleph_0$, but I could not prove that such $H$ does not exist (I think it does ...
0
votes
1answer
38 views
Finding the image of a homomorphism
I'm struggling a bit with how to find the image of a homomorphism. For instance, I'm given that
$$f:G\to G/H \text{ defined by } f(g)=gH \text{ is a homomorphism}$$
and I'm asked to prove that the ...
4
votes
2answers
62 views
Finding the kernel of a homomorphism
I have the groups of nonzero complex numbers and the positive real numbers and the homomorphism $f: \Bbb{C}^{*} \to \Bbb{R}_+$ such that $f(z)= \lvert z \rvert$. I need to find the kernel of f.
...
2
votes
4answers
72 views
number of subgroups of order $4$ of $\mathbb Z_4\oplus\mathbb Z_2?$
Without using the property of finite abelian group how to evaluate the number of subgroups of order $4$ of $\mathbb Z_4\oplus\mathbb Z_2?$
Please help ! I can show that $\mathbb Z_4\oplus\mathbb Z_2$ ...
5
votes
2answers
43 views
Let G be a group and let $a\in G$. Define $F_a:G\mapsto G$ via $F_a(x)=axa^{-1}$ for all $a\in G$. Prove that $F_a$ is an isomorphism from G onto G
This is what I have for my proof:
$F_{a^{-1}}(F_{a}(x))=F_{a^{-1}}(axa^{-1})=a^{-1}axa^{-1}a=(a^{-1}a)x(a^{-1})(a)=1x1=x$
...
-2
votes
1answer
159 views
Help with abstract algebra
Let $G=\{1,-1,i,-i,j,-j,k,-k\}$ where $i^2 =j^2 =k^2 =-1$, $-i=(-1)i,$ $1^2 =(-1)^2 =1$, $ij=-ji=k$, $jk=-kj=i$, and $ki=-ik=j$.
a) Construct the Cayley table for $G$
b) Show that $H=\{-1,1\}$ is ...
1
vote
2answers
48 views
Prove or Disprove the following: If $K$ is a maximal subgroup that is normal, in $G$ then $G \cong K \times_{\theta}H$
I think the statement is untrue. And I'm thinking that it can be disproved using a counterexample with the quaternion group and its maximal subgroups $\{1, -1, i, -i\}$, which are obviously normal. ...
1
vote
1answer
32 views
Studying the action of $GL(V)$ on the vector space $V$
The statement I am trying to prove is the following.
Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let
$\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural
...
2
votes
3answers
88 views
Homework - Prove that a given set is a group
I have a homework question and I don't know how to approach this exercise.
The exercise is the following:
Let's suppose $G$ is a set with binary function * defined for its members, which is:
...
1
vote
1answer
26 views
Groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.
I cant solve this exercise.
Find all groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.
I need a little help here. thanks!!!
10
votes
3answers
248 views
Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?
My guess is no. Does the first isomorphism theorem have anything to do with this?
Any hints appreciated, thanks.
1
vote
1answer
62 views
prove that $U_{51}$,$U_{80}$ are not isomorphic
I need prove the next result:
$U(\mathbb{Z}/51\mathbb{Z})$,$U(\mathbb{Z}/80\mathbb{Z})$ are not isomorphic.
thanks for your help!
1
vote
1answer
48 views
Groups of units: Find an explicit isomorphism $U_{35}$, $U_{39}$
I need help in the following exercise:
Find an explicit isomorphism between $U(\mathbb{Z}/35\mathbb{Z})$ and $U(\mathbb{Z}/39\mathbb{Z})$.
Thanks!
2
votes
2answers
44 views
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
I am trying to prove the following statement:
Let $G$ be a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K =
\{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times
K$.
...
7
votes
2answers
188 views
Abelian group admitting a surjective homomorphism onto an infinite cyclic group
I am working on the following problem:
Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective
homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$
By means of the First Isomorphism ...
1
vote
0answers
21 views
Tensor product of an irreducible $G$-representation and a one-dimensional representation [duplicate]
If $G$ is a finite group, $V$ is an irreducible $G$-representation and $W$ is any 1-dimensional $G$-representation (both over an algebraically closed field of characteristic zero), show that $V ...
1
vote
5answers
66 views
Show that If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$
If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$.
I am sort of stuck with this proof and I would appreciate a hint (not a full solution, please!). Preferably, ...
1
vote
2answers
39 views
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$.
i) Show that for every $m\in M$ and for every $n\in N$, $mn=nm$.
ii) If $G$ is generated by $M\cup N$ then $G\cong M ...
1
vote
1answer
74 views
Group action on subsets
Let $G$ be a group. Let $S$ be a subset of $G$ that is NOT a subgroup. Let $a,b$ be elements of $G$.
If $aS = bS$, must $a = b$?
The actual question is:
Let the dihedral group $D_3$ act on the set ...
0
votes
2answers
52 views
Finite group with a normal Sylow subgroup
Let $G$ be a finite group such that it has a normal Sylow p-subgroup. Is there any non-trivial element in the center of $G$?
1
vote
1answer
8 views
Showing a map multiplies its argument by some element on the right
Let $G$ be a group and define
$$\ell_g : G \to G, \qquad \ell_g(x)=gx,$$
$$r_g : G \to G, \qquad r_g(x)=xg.$$
Let $\phi : G \to G$ be a bijection such that $\ell_g \circ \phi = \phi \circ \ell_g$ ...
0
votes
1answer
47 views
show that $A \cap B$ is a normal subgroup of $B$ [duplicate]
Suppose $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$. Please help me to show that $A \cap B$ is a normal subgroup of $B$.
I know that since $B$ is subgroup of $G$, it has identity ...
1
vote
3answers
84 views
If $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$ , then $A\cap B$ is a normal subgroup of $B$
Suppose $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$. Show that $A\cap B$ is a normal subgroup of $B$.
7
votes
2answers
63 views
Complex finite dimensional irreducible representation of abelian group
I'm supposed to show that each Complex finite dimensional irreducible representation of an abelian group is one dimensional.
For any map $\phi: V \rightarrow V$ it holds that $\phi(\rho(g)v) = ...
3
votes
1answer
25 views
Show that the orbits of $S_n$ under the conjugation action of $S_n$ on itself correspond 1-1 with the cycle types.
Show that the orbits of $S_n$ under the conjugation action of $S_n$
on itself correspond 1-1 with the cycle types.
So, the orbit of $\sigma \in S_n$ is the set $S_n \sigma = \{ \tau .\sigma : ...
3
votes
4answers
58 views
Showing $(NM)/M \cong N/(N\cap M)$ for $N,M \triangleleft G$
This is problem 2.7 #6 from the second edition of Herstein's Topics in Algebra.
If $N,M$ are normal subgroups of $G$, prove that $(NM)/M \cong N/(N\cap M)$.
Any hints in the right direction?
0
votes
2answers
122 views
Suppose $N\leq H\leq G$, and $N\lhd G$. Prove $H\lhd G\iff H/N\lhd G/N$
Let N be a normal subgroup of $G$ and let $H$ be a subgroup of $G$. If N
is a subgroup of $H$, prove that $H/N$ is a normal subgroup of $G/N$ iff $H $ is
a normal subgroup of $G$
1
vote
0answers
35 views
Finite subgroups. [duplicate]
(Index of a subgroup $H$ in a group $G$ is the number of distinct left cosets of
$H$ in $G$, it is denoted as $|G : H|$)
How to solve that problem.
Let $H$ and $K$ be subgroups of a finite group ...
3
votes
2answers
52 views
Finite group cyclic [duplicate]
Let $G$ be a finite group. If for each m$\in$N, $x^m=e$ has at most $m$ solutions in $G$, $G$ is cyclic.
Can you give me a hint of this problem? I don't know.
