Cosets

Lagrange's Theorem The most important single theorem in group theory. It helps answer: How large is the symmetry group of a volleyball? A soccer ball? How many groups of order 2p where p is prime? (4, 6, 10, 14, 22, 26, …) Is 2257-1 prime? Is computer security possible? etc. To prove Lagrange's theorem, you need to understand cosets.

Cosets Definition: Let H ≤ G and a in G. The left coset of H containing a is the set aH = {ah | h in H} The right coset of H containing a is the set Ha = {ha | h in H} In additive groups use a+H and H+a. a is called the coset representative of aH. Similarly, aHa-1 ={aha-1 | h in H}

Cosets of K = {R0, V} in D4 Some left cosets of K: DK = {DR0, DV} = {D,R90} R180K = {R180R0, R180V} = {R180, H} Some right cosets of K: KD = {R0D, VD} = {D, R270} KR180 = {R0R180,VR180} = {R180, H} Questions: Is the left coset aK equal to the right coset Ka? Sometimes No Is the element a always a member of aK? Yes, because the identity is always in the subgroup K Is a coset a subgroup? No.

Cosets of K = {R0, V} in D4 Left Cosets R0K = {R0, V} = VK R90K = {R90, D} = DK R180K = {R180, H} = HK R270K = {R270, D'}= D'K Right Cosets KR0 = {R0, V} = KV KR90 = {R90, D'} = KD' KR180 = {R180, H} = KH KR270 = {R270, D} = KD How many elements in each coset of K? How many cosets of K are there? Howe many elements in the group D4?

More cosets Vectors under addition are a group: (a,b) + (c,d) = (a+c,b+d) Identity is (0,0) Inverse of (a,b) is (-a,-b) Associativity is easy to verify. H = {(2t,t) | t in R} is a subgroup. Proof: (2a,a) - (2b,b) = (2(a-b),a-b)

Visualizing H={(2t,t)} Let x = 2t, y = t Eliminate t: y = x/2 H

Cosets of H={(2t,t) | t in R} (a,b) + H = {(a+2t,b+t)} Set x = a+2t, y = b+t and eliminate t: y = b + (x-a)/2 The subgroup H is the line y = x/2. The cosets are lines parallel to y = x/2 !

H and some cosets (0,1) + H H (–3,0)+H (1,0) + H

Left Cosets of <(123)> in A4 Let H = <(123)> {, (123), (132)} H = {, (123), (132)} (12)(34)H = {(12)(34), (243), (143)} (13)(24)H = {(13)(24), (142), (234)} (14)(23)H = {(14)(23), (134), (124)}

Properties of Cosets: Let H be a subgroup of G, and a,b in G. 1. a belongs to aH 2. aH = H iff a belongs to H 3. aH = bH iff a belongs to bH 4. aH and bH are either equal or disjoint 5. aH = bH iff a-1b belongs to H 6. |aH| = |bH| 7. aH = Ha iff H = aHa-1 8. aH ≤ G iff a belongs to H

1. a belongs to aH Proof: a = ae belongs to aH.

2. aH=H iff a in H Proof: (=>) Given aH = H. By (1), a is in aH = H. (<=) Given a belongs to H. Then (i) aH is contained in H by closure. (ii) Choose any h in H. Note that a-1 is in H since a is. Let b = a-1h. Note that b is in H. So h = (aa-1)h = a(a-1h) = ab is in aH It follows that H is contained in aH By (i) and (ii), aH = H

3. aH = bH iff a in bH Proof: (=>) Suppose aH = bH. Then a = ae in aH = bH. (<=) Suppose a is in bH. Then a = bh for some h in H. so aH = (bh)H = b(hH) = bH by (2).

4. aH and bH are either disjoint or equal. Proof: Suppose aH and bH are not disjoint. Say x is in the intersection of aH and bH. Then aH = xH = bH by (3). Consequently, aH and bH are either disjoint or equal, as required.

5. aH = bH iff a-1b in H Proof: aH = bH <=> b in aH by (3) <=> b = ah for some h in H <=> a-1b = h for some h in H <=> a-1b in H

6. |aH| = |bH| Proof: Let ø: aH –>bH be given by ø(ah) = bh for all h in H. We claim ø is one to one and onto. If ø(ah1) = ø(ah2), then bh1 = bh2 so h1 = h2. Therefore ah1 = ah2. Hence ø is one-to-one. ø is clearly onto. It follows that |aH| = |bH| as required.

aH = Ha iff H = aHa-1 Proof: aH = Ha <=> each ah = h'a for some h' in H <=> aha-1 = h' for some h' in H <=> H = aHa-1.

aH≤G iff a in H Proof: (=>) Suppose aH ≤ G. Then e in aH. But e in eH, so eH and aH are not disjoint. By (4), aH = eH = H. (<=) Suppose a in H. Then aH = H ≤ G.