Top Posts Tagged with #group-theory

Exceptional Automorphisms of S_6

The symmetric group S_6 has a special property that S_n does not have for ANY n ≠6. Really? 6, of all numbers?? How odd.

For any group G, and every g in G, the conjugation map f_g: h ↦ g h g^-1 is an automorphism of G. That is, it is an isomorphism from G -> G. The group of these particular automorphisms (under composition) is called the group of inner automorphisms of a group. Did you know that for every symmetric group S_n EXCEPT n=6, this is the entire automorphism group?

For some reason, S_6 has basically ONE (and only one) weird automorphism. What I mean by this is if we denote the inner automorphisms of S_6 by Inn(S_6), then S_6/Inn(S_6) ~= Z_2. (In other words, you can pick a single automorphism sigma such that the automorphisms of S_6 that are not inner automorphisms can be written as a composition of inner automorphisms and this one sigma.)

What is also interesting is the proofs that there are no outer automorphisms for n≠6. Basically you can eliminate all but finitely many n, then you can pick off cases, and you are left with an n=6 shaped hole in your cases that CANNOT be filled, which feels so weird to me. If I was proving this myself I'd be going crazy having proven it for every case except 6, and having to resort to some proof which for some reason doesn't work for finitely many values.

Construction

There are so many different ways to construct the weird automorphism of S_6 - I have some links at the bottom. I particularly like the graph theory/geometric ones. Something about using factorisations of K_6 or a dodecahedron just makes the 6 feel more unique. I will admit I don't think I understand what is fundamentally special about 6 enough yet, though, on a philosophical level.

Practically, though, the constructions all basically boil down to the fact you can put a copy of S_5 inside S_6 in a way that isn't the obvious way, which you can only do for n=6. That is the special part.

After that, you have these 6 cosets of S_5 inside S_6 that S_6 acts upon. In other words, each element of S_6 permutes the 6 cosets of S_5 living inside it. But the group of permutations of 6 cosets is S_6. So we have a mapping from elements of S_6 to S_6 - an automorphism! Is this automorphism an inner automorphism? Each construction shows why they are outer differently, but a common theme is to show that the mapping S_6 to S_6 does not take transpositions to transpositions, but inner automorphisms preserve the cycle type.

Some proofs and examples

The wikipedia article. I like the construction about graph partitions. It does not, however, have much detail sadly.

Fortunately, the graph partitions thing from wikipedia is explained here. It's very short and to the point, also quite nice to look at:

Let’s consider the group S6 of permutations of {1,2,3,4,5,6}. For fairly boring reasons, we can find S5 living inside there — the subgroup w

Fairly elementary explanations, followed by more intense ones:

Requires only basic group theory to understand the first few explanations they provide (although it isn't trivial if you just learned group theory). Bonus points for "MyStIc PeNtAgOnS" (capitalisation mine):

We use a simple description of the outer automorphism of S_6 to cleanly describe the invariant theory of six points in P^1, P^2, and P^3.

Allegedly useful, but I can't grab a copy: (I think its on mathscinet under MR1240362 as per David Leep's personal website's publications section)

Combinatorial Structure of the automorphism group of S6 by T.Y. Lam and David B. Leep, Expositiones Mathematicae11 (1993), no. 4, pp. 289-903.

The comment by Matthew Towers here is also interesting:

$$ \begin{array}{|l|c|c|} \hline \text{cycle structure} & \text{number of permutations} & \text{order} \\ \hline 6 & 120 &a

#group-theory #math #maths #mathematics #automorphisms #MyStIcPeNtAgOnS

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群論のメモ

準同型定理に至るまでにおさえておきたい事項。準同型写像の核と正規部分群の関係、自然な準同型。準同型写像全体のなす群を考え、自己同型群と内部同型の関係を調べると、これもまた正規部分群になっているね、すっごーいたーのしー！みたいなメモ。

#group-theory #math

Applying Sylow theorem for finding Normal Subgroup

Sylow Theorem: Let |G|=p^{\alpha}m, for some prime p such that p^{\alpha +1} \nmid |G|. If n be the number of p- Sylow subgroup(subgroup of order p^{\alpha}), then it will satisfy the following condition

n \mid m, and

n = 1 ~(\!\!\!\mod p).

Any subgroup of unique order is normal.

Classification of group of order pq: If G is a group of order pq, for primes p<q then we can have following possibilities

If p \nmid (q-1), then only cyclic group of order pq is possible.

If p \mid (q-1), then we get one cyclic group of order pq and one non-abelian group of order pq

G is a finite group of order p^2q where in p and q are distinct primes such that q \nmid (p^2-1) and p \nmid (q-1). Then G is an abelian group.

#Group-Theory #Sylow-Theorem

How to write class equation of Dihedral Group

Class equation of dihedral group, D_{2n}, is given by,

when n is odd, i.e., n =2m+1, 2 n=1 + \underbrace{2 +2+ \cdots +2}_{m \rm{\,times}} +n,

when n is even, n = 2m, 2n = 1+1+ \underbrace {2 +2 +\cdots +2 }_{m-1 ~\rm{ times}} +m +m.

#Group-Theory

Permutation Group S3 and S4.

Elements of S_3

The symmetric group on three letters S_3 has 6 elements.

Cycle Structure Elements Even/Odd Identity (~) Even All two cycle. (12),(13),(23) Odd All three cycle (123),(132) Even

Note: The 3 order subgroup generated by (123) is a normal subgroup of S_3 and denoted by A_3.

Elements of S_4

The symmetric group on three letters S_3 has 6 elements.

Cycle Structure Elements Even/Odd Identity (~) Even All 2-cycles (12),(13),(14),(23),(24),(34) Odd Product of two 2-cycles (12)(34),(13)(24),(14)(23) Even All 3-cycle (123),(132),(124),(142),(134),(143),(234),(243) Even All 4-cycle (1234),(1432),(1324),(1423),(1243),(1342) Odd

Note 1: All even elements of S_4 form a normal subgroup of order 12 is known as A_4.

Note 2: Product of disjoint two 2-cycle with identity forms a normal subgroup of order 4 in S_4. It will be automatically normal in A_4.

#Group-Theory #Permutation

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The 5/8 bound

Suppose $ G $ is a nonabelian finite group. A fun question to ask is is `what is the probability that two elements of $ G $, selected at random with replacement, commute?' Let's denote this probability by $\Pr(G)$. A nice folklore result, first recorded (I think) by Gustafson in a note in 1973, says: Theorem: $\Pr(G) \leq \frac{5}{8}$. To prove this we can use the following fundamental fact: Lemma: $\Pr(G) = \frac{k(G)}{|G|}$ where $k(G)$ is the number of conjugacy classes in $G$. Proof: First, note that \[ \Pr(G) = \frac{|\{(x,y) \mid x,y \in G, xy=yx \}|}{|G|^2}. \] Let \[ C = \{ (a,b) \in G \times G \mid ab=ba \}.\] If we fix $x \in G$, then the number of elements in $C$ of the form $(x,y)$ is $|C_G(x)|$, where $C_G(x)$ is the centraliser of the element $x$ in $G$. So, \[ |C| = \sum_{x \in G} |C_G(x)| .\] Now, let $x_1, \ldots, x_k$ be representatives of the $k=k(G)$ conjugacy classes in $G$. If $x$ is conjugate to $y$ in $G$, then $C_G(x)$ and $C_G(y)$ are conjugate subgroups and so $|C_G(x)|=|C_G(y)|$. So, since $|x^G| = |G : C_G(x)|$, we have \[ \begin{align*} |C| &= \sum_{i=1}^k |x_i^G| \cdot |C_G(x_i)| \\\\ &= \sum_{i=1}^k |G : C_G(x_i)| \cdot |C_G(x_i)|\\\\ &= \sum_{i=1}^k \frac{|G|}{|C_G(x_i)|} \cdot |C_G(x_i)| = k \cdot |G|. \end{align*} \] So, \[ \Pr(G) = \frac{|C|}{|G|^2} = \frac{k(G) \cdot |G|}{|G|^2} = \frac{k(G)}{|G|}. \] Now, to prove Gustafson's result, note that the class equation says that \[ |G| = |Z| + |K_1| + \cdots + |K_t| \] where $|Z|$ is the centre of $G$ and the $K_i$ are the nontrivial conjugacy classes. So, $|K_i| \geq 2 $ and so \[ \frac{|G| - |Z|}{2} \geq t \] i.e. \[ k = t + |Z| \leq \frac{|G| - |Z|}{2}. \] Since $G$ is nonabelian, $G/Z$ is noncyclic and so $ |Z| \leq \frac{|G|}{4}$. So, $ k \leq \frac{5}{8}|G|$ and so $\Pr(G) \leq \frac{5}{8}$. The bound is realised by the dihedral group of order 8 and so is best-possible. In the same note, Gustafson notes that if $G$ is a (nonabelian) compact, Hausdorff topological group then the same bound holds (where the probability computations are done via the unique normalised Haar measure on $G$). It is much harder to find good lower bounds for $\Pr(G)$, although Gustafson notes that Erdös and Turán (in a paper in 1968 which is, to the best of my knowledge, the first place anybody considers $\Pr(G)$) prove that if $G$ is a finite group, then \[ \Pr(G) \geq \frac{\log_2 \log_2 |G|}{|G|} .\]

#mathematics #group-theory

Automorphism towers

A favourite theorem: Let $ G $ be a group, let $ \mathrm{Aut}(G) $ denote the full automorphism group of $ G $ and let $ Z(G) $ denote the centre of $ G $. If $ G $ is a centreless group, that is if $ Z(G) = 1 $ then it is easy to show that $ \mathrm{Aut}(G) $ is also a centreless group and, since $ G / Z(G) \cong \mathrm{Aut}(G) / \mathrm{Inn}(G) $, where $ \mathrm{Inn}(G) $ denotes the group of inner automorphisms of $G$, we see that a centreless group $G$ embeds in its automorphism group. So, we can identify $G$ with $\mathrm{Inn}(G)$ via the natural embedding. Let $G_0 = G$ be a centreless group and, for each ordinal $\alpha$, let \[G_{\alpha+1} = \mathrm{Aut}(G_{\alpha}) \] or, if $\alpha$ is a limit ordinal, let \[ G_{\alpha} = \bigcup_{\beta \lt \alpha} G_{\beta} . \] Each of the $G_i$ is centreless and, by making suitable identifications, we obtain an ascending chain of groups \[ G_0 \unlhd G_1 \unlhd G_2 \unlhd \cdots \unlhd G_{\alpha} \unlhd G_{\alpha+1} \unlhd \cdots .\] This is the automorphism tower of the group $G$. The natural question that arises is: ``does the automorphism of G terminate?" A 1939 result of Wielandt, the proof of which is rather technical, says that if $G$ is a finite centreless group, then the automorphism tower of $G$ terminates after finitely many steps. The problem remained open for infinite centreless group (some progress was made in special cases) until, in 1985 in a wonderful 3-page paper, Thomas proved: Theorem: The automorphism tower of any centreless group terminates after at most $ (2^{|G|})^+$ steps (where, if $\lambda$ is a cardinal, then $\lambda^+$ is the successor cardinal of $\lambda$). In 1998, in another 3-page paper, Hamkins proved that the automorphism tower (suitably defined) of any group terminates. More information may be found in this MathOverflow discussion and in the excellent (free!) book of Simon Thomas.

#mathematics #group-theory

Exceptional Automorphisms of S_6

The symmetric group S_6 has a special property that S_n does not have for ANY n ≠6. Really? 6, of all numbers?? How odd.

Construction

Some proofs and examples

The wikipedia article. I like the construction about graph partitions. It does not, however, have much detail sadly.

Fortunately, the graph partitions thing from wikipedia is explained here. It's very short and to the point, also quite nice to look at:

Let’s consider the group S6 of permutations of {1,2,3,4,5,6}. For fairly boring reasons, we can find S5 living inside there — the subgroup w

Fairly elementary explanations, followed by more intense ones:

We use a simple description of the outer automorphism of S_6 to cleanly describe the invariant theory of six points in P^1, P^2, and P^3.

Allegedly useful, but I can't grab a copy: (I think its on mathscinet under MR1240362 as per David Leep's personal website's publications section)

Combinatorial Structure of the automorphism group of S6 by T.Y. Lam and David B. Leep, Expositiones Mathematicae11 (1993), no. 4, pp. 289-903.

The comment by Matthew Towers here is also interesting:

$$ \begin{array}{|l|c|c|} \hline \text{cycle structure} & \text{number of permutations} & \text{order} \\ \hline 6 & 120 &a

#group-theory #math #maths #mathematics #automorphisms #MyStIcPeNtAgOnS

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群論のメモ

#group-theory #math

Applying Sylow theorem for finding Normal Subgroup

n \mid m, and

n = 1 ~(\!\!\!\mod p).

Any subgroup of unique order is normal.

Classification of group of order pq: If G is a group of order pq, for primes p<q then we can have following possibilities

If p \nmid (q-1), then only cyclic group of order pq is possible.

If p \mid (q-1), then we get one cyclic group of order pq and one non-abelian group of order pq

G is a finite group of order p^2q where in p and q are distinct primes such that q \nmid (p^2-1) and p \nmid (q-1). Then G is an abelian group.

#Group-Theory #Sylow-Theorem

How to write class equation of Dihedral Group

Class equation of dihedral group, D_{2n}, is given by,

when n is odd, i.e., n =2m+1, 2 n=1 + \underbrace{2 +2+ \cdots +2}_{m \rm{\,times}} +n,

when n is even, n = 2m, 2n = 1+1+ \underbrace {2 +2 +\cdots +2 }_{m-1 ~\rm{ times}} +m +m.

#Group-Theory

Permutation Group S3 and S4.

Elements of S_3

The symmetric group on three letters S_3 has 6 elements.

Cycle Structure Elements Even/Odd Identity (~) Even All two cycle. (12),(13),(23) Odd All three cycle (123),(132) Even

Note: The 3 order subgroup generated by (123) is a normal subgroup of S_3 and denoted by A_3.

Elements of S_4

The symmetric group on three letters S_3 has 6 elements.

Note 1: All even elements of S_4 form a normal subgroup of order 12 is known as A_4.

Note 2: Product of disjoint two 2-cycle with identity forms a normal subgroup of order 4 in S_4. It will be automatically normal in A_4.

#Group-Theory #Permutation

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Watch Anya Live on Cam

Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.

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The 5/8 bound

#mathematics #group-theory

Automorphism towers

#mathematics #group-theory

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