\(\def\ < ={ \leq }\)
\(\def\ > ={ \geq }\)
1.6 Symmetric, Alternating, and Dihedral Groups
- DEF Even/Odd permutation is written in even/odd number of transpositions
- THM A permutation can be just 1 of even or odd
- DEF \(A_n\) set of all even permutations of \(S_n\) where \(n > 2\)
- SUB TOPIC FOCUSED ON TRANSPOSITIONS
- PRP interesting property of transpositions
- (bx)(ab) = (ab)(ax) [flipping]
- (bx)(ab)(bx) = (ab)(ax)(bx) = (ab)(xab) = (ab)(abx) = (ax) [replacing a number in transposition form xtx]
- (cb) (ab)(ac)(ab)(ac) (cb) = (ab)(ac)(ac)(ab)(ab)(ac) = (ab)(ac)
- (ab) (ab)(ac)(ab)(ac) (ab) = (ac)(bc) = (cab)
- (ac)(ab)(ac) (ab) = (ab)(bc) (cb)(ac) = (abc)
- (ab)(ac)(ab)(ac) = (ac)(bc)(cb)(ac) = (acb)
*(ac)(cb)(bc)(ab) = (ac)(ab)
- THM a 2-cycle in normal subgroup generates all 2-cycles
- Let (ab) our 2 cycle, then multi by
- (bx)(ab)(bx) = (ax)
- THM a 3-cycle in normal subgroup generates all 3-cycles
- Let (abc) be our 3 cycle (ab)(ac)
- (cx)(abc)(cx) = (cx)(ab)(ac)(cx) = (ab)(cx)(ac)(cx) = (ab)(ac)(ax)(cx) = (abc)(xac) is in N
- \((abc)^{-1} \in N\) so \((abc)^{-1} (abc)(xac) = (xac) \in N\)
- Carefully consider ALL the possibilties of the objects/properties of the objects you are working with.
- Once you have exhausted, you can explore including additional property/structure/restriction
- THM \(A_n\) is subgroup of \(S_n\) of index 2 with order \(|S_n|/2\). \(A_n\) is the only subgroup of \(S_n\) of index 2.
- index of 2 implies normal
- index 2? is there some way to characterize even vs odd permuation? how about start with small case were we can definitely know n=2
- we have I and (1,2). Now, lets extend this to n=3. How do we for all permutations of \(S_3\) from \(S_2\)?
- \(|S_2|=2\) \(|S_3|=3*2\)
- Alternate notation for \(S_2\): [1,2] [2,1]
- A let's try to show a 3 cycle exists
- well, \(A_n\) cannot consist of all odd cycles, whose size will have index 2,
- because it would contain all 1 cycles that generates all of \(S_n\)
- so it contains at least 1 even cycle
- B can we reduce this even permu into a 3 cycle?
- let us consider the disjoint cycles of thie even permu
- A.1 could we say it contains n cycle then it must contain n-1 cycle?
- N! permutations, N!/2
- this is just reordering the cycle (ab)(ac)(ad) = (ad)(ab)(ac)(ad)(ad) = (ad)(ab)(ac)
- this property is interesting (ab)(ac)(ad)(ac)(ad)(bc) = (ab) (adc) (bc) = (ab)(ad)(ac)(bc)
- let me write a program to do these computations
- (dx)(abcd)(dx) = (dx)(ab)(ac)(ad)(dx) = (ab)(ac)(ad)(ax)(dx) = (ab)(ac)(ad)(xad) = (abcd)(xad)
- C \((abcd)^{-1}\) is in our set so (xad) is in our set
- we can do this with any n-cycle; it's existance in \(A_n\) means a 3-cycle exists in \(A_n\)
- Let's look at for 2-cycle (cx)(ab)(ac)(cx) = (ab)(ac)(ax)(cx) = (ab)(ac)(ax)(cx) -> (ax)(cx) (this is for showing that any 3cycle exists)
- C.1 well, if we have disjoint cycle, could we not modify the later cycle to be whatever we want? and then we can multi by inverse
- (cx)(cycle1)(abc)(cx) = (cycle1)(cx)(abc)(cx) = (cycle1)(abc)(ax)(cx) => (ax)(cx) in our set bc, \(((cycle1)(abc))^-1\) in our set
- D so if we show that a cycle exists in \(A_n\)
- how many cycles are there? (N)(N-1)/2 + (N)(N-1)(N-2)/3 + (N)*...(1)/N
- how many permutations are there N!. what about |A_n| = N!/2
- Counter example. N=8, |cycles|<|even permu|
- DEF Simple group: a group without proper normal subgroups
- only simple abelian groups are \(Z_p\)
THM Approx wording: A_n is simple for n!=4
- suppose there is normal subgroup of \(A_n\), \(C\)
- take any element of c, any permutation can be decomposed into disjoint cycles
- a 3-cycle in a normal subgroup generates all of it
- \(A_2\) is simple because it is just \({1}\)
- \(A_3\) is simple because it is just consists of 3-cyles and any subgroup contains a 3-cycle
- \(A_4\) is not simple because ... of the cyclic subgroup generated by (1234)
- if for \(A_5\) we have access to 5 variables, allowing us to form the below equation
- since C is normal, the below equation is also in C
- (cx)(cycle1)(abc)(cx) = (cycle1)(cx)(abc)(cx) = (cycle1)(abc)(ax)(cx) => (ax)(cx) in our set bc, \(((cycle1)(abc))^-1\) in our set
LEM if a normal subgroup contains 3-cycle, then it generates all of \(A_n\)
- The book's proof 6.11 and 6.12 I don't really like
- DEF Dihedral Groups: \(D_n\), subgroup of \(S_n\) generated by
- a = (12...n)
- b = (1, 2, 3 , ... n-1,n)
- ----(1, n,n-1, ... 3 ,2)
- = (2,n)(3,n-1)(4,n-2)...
- ???
- THM What is \(|D_n|\)
- elements are of form \(a^m b^n\), \(a^m b^n\)
- we would like to know which ones are equal to each other and find the unique ones
- A The \(a^i b^k\), \(a^j b^k\) i,j \(a^i=a^j\) -> i=j
- B How about these forms? \(a^i b = a^j\) then \(b = a^{j+n-i}\) well the only form of \(a^n\) that preserves 1->1 is the identity which is clearly not b
- C How about \(a^i b = b a^j\)?
- lets look at \(ab = b?\)
- a *[b]
- (12)(13)(14)...(1n) * [(2,n)(3,n-1)(4,n-2)...]
- [(2,n)(3,n-1)(4,n-2)...] * ...(1,4)(1,3)(1,2)
- this is because the transpositions are disjoint except for 1 element in [(2,n)(3,n-1),...] that causes the swap
- the second term is precisely \(a^{-1} = a^{n-1}\)
- Let use \(ab = ba^{-1}\) -> \(a^i b = b a^{-i} = b a^{n-i}\)
- this means \(\{a^i b\} \subset \{b a^j\}\) and vice-versa -> \(\{a^i b\} = \{b a^j\}\)
- Theorem describes generator
Questions
Discourse in Exercises
Generator of \(S_n\) is \((12), (123...n)\)
- Generator of \(S_n\) is \((12),(13),...(1n)\)
- Generator of \(S_n\) is \((12),(23),(34),...,(n-1 n)\)
- Generator of \(S_n\) is \((1234..n) (12)\)
- Interesting property
- \(s(abcd)s^-1 = (s(a),s(b),s(c),s(d))\)
\(a < = b\) and \(b < = a\) \(\implies\)
- you could look \( < =\) as \( < \) or \(=\); and the two statements rule out the possibility of \( < \)
- but a similar statement with sets
\(F = s o s^-1\) , \(s\) permutation , \(o = (abcd)\), cycle
- \(F(s(a))\)
- \(= s(o( s^{-1}(s(a)) ))\)
- \(= s(o( a ))\)
- \(= s(b)\)
- \(= (s(a), s(b))\)