New answers tagged group-theory
1
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Let a group $G=\left\{\begin{pmatrix}a&b\\0&a\end{pmatrix} : a,b\in\mathbb{Z_3}, a\neq{0}\right\}$. Find the order of $G$ and find all subgroups
Let
$$X=\begin{pmatrix}
2 & 1 \\
0 & 2
\end{pmatrix}.$$
Observe that $|X|=6$ and $X$ generates the group $G$. Hence $G\cong \Bbb Z_6$.
Theorem: Let $G$ be a finite, cyclic group. Suppose $d\...
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Let a group $G=\left\{\begin{pmatrix}a&b\\0&a\end{pmatrix} : a,b\in\mathbb{Z_3}, a\neq{0}\right\}$. Find the order of $G$ and find all subgroups
We can easily see that the elements of the group $G$ are $\{ x_1 = \left(\begin{matrix}1 & 0\\0 & 1\end{matrix}\right), x_2 = \left(\begin{matrix}1 & 1\\0 & 1\end{matrix}\right), x_3 = ...
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2
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Order of a product of two elements in a Frobenius Group
By a straightforward counting argument, a Frobenius group of degree $n$ has exactly $n-1$ fixed-point-free elements, so these must be the non-trivial elements of $K$.
So all elements of $G$ outside of ...
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The number of the double cosets $K\backslash G/H$ equals $ \langle \pi_{G/K} , \pi_{G/H} \rangle$
Since we are finding the number of the $K$-orbits, then
\begin{equation}
\langle Res_K^G(\pi_{G/H}) , 1_K \rangle_K = \langle \pi_{G/H}, Ind_{K}^{G} 1_K\rangle_G = \langle \pi_{G/H}, \pi_{G/K} \...
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In SageMath, how to put group element into standard form
OK, I have found that I can do it with the following code
print(G[G.list().index(q*w)])
This will print a^-3 rather than ...
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Number of non-isomorphic abelian groups of order $1440$, containing an element of order $180$.
A different formulation of the Structure Theorem of finite abelian groups states that every such group can uniquely be written in the form $$\bigoplus_{p\text{ prime}}\bigoplus_{i=1}^{r_p}\mathbb Z/p^{...
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Normality of the lower central series
We will try to prove that $\gamma_n(G) \lhd G$ by induction. This is clearly true for $n=1$ as $\gamma_1(G)=G \lhd G$.
Now suppose we know that $\gamma_n(G) \lhd G$.
Now consider $\gamma_{n+1}(G)$. It ...
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The number of the double cosets $K\backslash G/H$ equals $ \langle \pi_{G/K} , \pi_{G/H} \rangle$
We have an isomorphism $\mathbb C[K\backslash G] \otimes_{\mathbb C[G]} \mathbb C[G/H]\cong \mathbb C[K\backslash G/H]$ given by $e_{Kg}\otimes e_{g'H}\mapsto e_{K(gg')H}$ Since $\dim \mathbb C[K\...
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Cataloging the Core-free Subgroups of the Permutation Group
For $n > 4$ the symmetric group $S_n$ has exactly three normal subgroups: $1$, $A_n$, $S_n$. See here for example. So a subgroup is core-free precisely when it doesn't contain $A_n$. Since $A_n$ is ...
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Showing that a group of order $231$ is a semi direct product
You say that you know there is exactly one Sylow-7-subgroup. Call it $X$, then $X$ is normal in $G$.
You say that you know there is exactly one Sylow-11-subgroup. Call it $Y$, then $Y$ is normal in $G$...
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Finite groups with elements of order a prime power
There is a complete classification of such groups, see Theorem 1.7 of this paper and this paper. These groups are called EPPO groups and are extensively used in graphs defined on groups.
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How fast is taking the quotient of a group by a subset?
It looks like this can be done in $O\big(|G|\cdot\log (|G|)\cdot \alpha(|G|)\big)$ time, where $\alpha$ is the very slow-growing inverse Ackermann function--an upper bound is $O(|G| \log^2 |G|)$ time.
...
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3
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Characterization of transpositions in symmetric group?
Here is an alternative way to characterize the transpositions, which works in $S_X$ whenever $|X|>6$: $\sigma$ is a transposition if and only if $\sigma$ has order $2$, and for every conjugate $\...
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2
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Canonical Identification of Tangent Vector of Matrix Lie Group
$\newcommand\R{\mathbb{R}}\newcommand\M{\mathcal{M}}$
MORE BRIEF COMMENTS:
A tangent vector at $M \in G$ is a matrix.
Example: Let $$G = \left\{ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\...
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Canonical Identification of Tangent Vector of Matrix Lie Group
In regard to some of your actual questions, The Lie algebra $\mathfrak{g}$ of some matrix Lie group $G$ is precisely equal to the tangent space at the identity with the addition of an appropriate ...
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-1
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Length of orbits in group actions
Maybe three years late, but we can get 5-orbits possibilities with GAP code:
Filtered(Partitions(67, 5), q -> ForAll(q, r -> r in DivisorsInt(40)));
which ...
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7
votes
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How to find the $\mathbb Q$ conjugates?
Let $K$ be the splitting field of the polynomial $$f(X) = X^6+12X^4-14X^3+48X^2+168X+113$$ over the field $\mathbb{Q}$.
We denote $\beta$ a $\mathbb{Q}$-conjugate of $\alpha = \sqrt[3]{7}+2i$ in $K$. ...
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On $\mathrm{Gal}(\mathbb{Q}(e^{\frac{2i\pi}{n}}/Q)$
There is no need to know the size of that Galois group to see it is abelian, and it is much simpler to think of it as the splitting field over $\mathbf Q$ of $x^n - 1$.
Let $\sigma$ be an automorphism ...
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On $\mathrm{Gal}(\mathbb{Q}(e^{\frac{2i\pi}{n}}/Q)$
For $\sigma$ in $G=Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ we have that $\sigma(\zeta_n)$ is an $n$-th primitive root of unity, so it can be written $\sigma(\zeta_n)=\zeta_n^k$ for some $k$ coprime to $...
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Reference Request: ADE Resolution and McKay Correspondence
Alex already gave a lot of good references. A brief introduction or the summary of the classical results can also be found here: https://cdn.jsdelivr.net/gh/Ju-Tan/Slide/McKay_Correspondence_v2.pdf .
...
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Visual intuition for normal subgroups
As someone who spent a few years trying to understand this part of algebra more visually I have two comments:
Equation (1) tells you nothing about the internal structure of $H$ -- if you use the ...
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Reference regarding quotients of link groups induced by link inclusions
When you add in the relator that sets a given meridian $=$ identity in the group (which has the effect of taking the quotient by the normal subgroup corresponding to that relator), this is the same as ...
- 446
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Clarification about the definition of word (group theory)
What might be confusing you is that we're working in $G$ not $S$. $S$ indeed doesn't have to contain the inverses of its elements - it's a subset, not a subgroup - but those inverses must exist ...
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Interpreting group in normal abelian subgroup of finite index?
I think the exercise is wrong, or at least that the method suggested in the hint doesn't work.
Let $A$ be an abelian group with an involution $\varphi\colon A\to A$ which is not definable. For example,...
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Clarification about the definition of word (group theory)
Not sure I'm understanding the question, but...if $s$ is an element of $S$, then it's also an element of $G$, which means that the group structure of $G$ gives us an inverse $s^{-1}$.
In the same way, ...
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Is the group $\langle x,y:x^5=y^2=1\rangle$ a quotient of some subgroup of $GL_2(\mathbb F)$?
$SL_2(\mathbb{Z})$ contains a free group of countable rank so any countable group is a quotient of a subgroup of $GL_2(\mathbb{Z})$.
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Orientation for groups
Here is an answer. I will add references later.
Let $G$ denote the group $PSL(2, \mathbb R)$; it is the group of orientation-preserving isometries of hyperbolic plane. The group $G$ is connected and ...
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4
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Turing degree of group isomorphism problem
The isomorphism problem for finitely presented groups is $\Sigma_1$-complete.
The proof of undecidability in fact encodes the halting problem: we produce, for each $k$, a pair of finitely presented ...
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Determining if a field extension of a function field is a splitting field
The only way we can characterize the solution of $X^3-t$ is "it is a zero of the polynomial $X^3-t$ in an algebraic closure of $\mathbb Z_3(t)$".
(Yes I admit this is abstract).
However $\...
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Interesting examples of lattice ordered abelian groups
This is less a direct answer than a hopefully-helpful comment, but: there is an analogue of Cayley's theorem for abelian $\mathscr{l}$-groups! Every abelian $\mathscr{l}$-group can be viewed as a ...
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Exercise 2.6 in Katok's book "Fuchsian Groups"
Here is a counter-example to the implication (iii)$\Rightarrow$(i). Consider $\mathbb R$ with the metric
$$
d(x,y)=\min\{ |x-y|, 1\}.
$$
Let $X=\mathbb R\times \{0,1\}$ denote the disjoint union of ...
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A confusion related to concrete category in Hungerford Book
(since morphisms are not functions on the set $G$)
This is a poor way to phrase it. Whether something is or isn't a function on $G$ is not an invariant statement under isomorphism, and thus violates ...
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$G$ is finite iff $G/N$ and $N$ are finite
The set $G$ is the union of all sets of the form $gN$, and there are only finitely many of them. Besides, each $gN$ has as many elements as $N$, which is finite. Therefore, $G$ is finite.
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Intuition cosets of $S_n$ under product of "subgroups"
Some general context first. The denominator is called a Young subgroup, and in this case it's a maximal parabolic subgroup. The quotients are basic building blocks in the representation theory of the ...
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What is the continuous product
See Product integral. Apparently there is not a single definition but several that follow the same idea.
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The center of $\langle x,y:x^3=y^2=1\rangle$
There is no reason why the intersection being trivial would imply a trivial center. In the dihedral group of order $8$, the group is generated by a rotation and a reflection, the cyclic subgroups they ...
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Intuition cosets of $S_n$ under product of "subgroups"
Let $G=S_n$, acting naturally on $X:=\{1,\ldots,n\}$ and $H=S_2\times S_{n-2}$ be the subgroup of $G$ that preserves $\{1,2\}$ setwise. Let $\binom{X}{2}$ be the set of unordered pairs of elements of $...
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What is the smallest finite group $G$ with a proper nontrivial noncentral subgroup $H$ to have a non-abelian centralizer $C_G(H) \neq G$?
The smallest example is $G=S_4$, of order $24$, with $H$ a cyclic group of order $2$ generated by a double transposition, and the centraliser of $H$ is dihedral of order $8$.
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finding presentation for a group quotient modulo normal cyclic subgroup
Either by definition or using the properties of the free group, we know that $G=\langle x,y\mid x^5=y^2\rangle$ is isomorphic to $F/M$, where $F$ is the free group of rank $2$ generated by $x$ and $y$,...
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Same conjugacy implies same isotropy?
As @art and @kan't point out, the even more general fact is that elements in the same orbit (which in the case of the conjugation action are conjugate elements) have conjugate isotropy groups.
Now ...
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Tangent space of Lie group SO(n)
The tangent space of a smooth manifold at a point is the set of all possible directions in which a curve can be drawn through that point. The tangent space of $SO(n)$ at a point $R$ is the set of all ...
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Direct product, semidirect product and associativity
I think there is no such law, and at the end of your post you already hinted at a counterexample.
In the alternating group $A_4$, the three double transpositions, together with the identity, form a ...
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Same conjugacy implies same isotropy?
In general, any $G$-action on any set $S$ induces an equivalence relation$^1$ on $S$ via:
$$t\sim s\iff \exists g\in G\mid t=g\cdot s$$
The equivalence class of $s\in S$ is then:
\begin{alignat}{1}
[s]...
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Inverse image of a cyclic normal tower is also cyclic and normal via canonical homomorphism
If $M\unlhd G/H$, denote by $\bar{M}=\Phi^{-1}(M)$. As you said, $\bar{M}\unlhd G$. Moreover, $H\leq \bar{M}$ and $\bar{M}/H\simeq M$ (see for example the answer of this: The correspondence theorem ...
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Same conjugacy implies same isotropy?
This is wrong and irrelevant. And Lang does not assume that "same conjugacy, same isotropy."
(They are called conjugacy classes, not "conjugate classes")
The correct statement is ...
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How to prove that Binary 1 is a cyclic group and Binary 2 is a dihedral group?
Here is how you can use Cayley table to verify that set G can represent both a cyclic group $C_4$ and a dihedral group $D_2$ which has order 4( 2 reflections and 2 rotations):
I have a set G ={e,a,b,c}...
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Choice of group action on Sylow subgroups
The kernel of your action is exactly the same as the kernel of the action on the Sylows by conjugation, and maps to the same symmetric group. So you are not getting "way more information about ...
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What are the automorphisms of finite commutative groups？
Yes, there is a complete answer to this question. See, for instance, https://arxiv.org/abs/math/0605185.
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Group theory orbit stabilizer
By the orbit-stabilizer theorem, all the pointwise stabilizers have order $\frac{|G|}{|X|}$. But then the union of all of them has size at most $|X|\Bigl(
\frac{|G|}{|X|}-1\Bigr)=$ $|G|-|X|<|G|$, ...
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Simple part in Butterfly lemma proof
Let $xy\in u(U\cap v)$ with $x\in u$ and $y\in U\cap v$.
If $z\in U\cap V$, then $zxz^{-1}\in u$ because $x\in u$, $u\triangleleft U$, and $z\in U$. Also, $zyz^{-1}\in U$ because $z,y\in U$. In ...
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