NFL Basic Structure& Lagrange's Theorem
The NFL has 32 teams, which serves as our "group G"
What is the total size (order) of the NFL group?
The order of U(10) equals the number of teams in half of a division.
What is 4?
The NFC South (4 teams) is a subgroup of the NFC (16). Lagrange’s theorem says the NFC is split into this many equal “parts.”
What are 4 cosets?
Each NFL conference has 16 teams, and this fact demonstrates a key divisibility property of subgroups.
What is an example of Lagrange’s theorem?
U(9) has 6 elements, which demonstrates this arithmetic function that counts numbers coprime to 9.
What is Euler's totient function?
In U(15), there are 8 elements. Subgroup orders must divide 8 by this theorem.
What is Lagrange’s theorem?
If each conference is 16 teams, explain why the NFL can’t form a subgroup/division of size 6.
What is because subgroup orders must divide 16 or 32 by Lagrange’s theorem, and 6 does not?
Compute all elements of U(16) and use them to argue why a subgroup of order 6 cannot exist inside it
What is because |U(16)| = 8, and subgroup orders must divide 8, so 6 is impossible?
NFL playoffs allow 7 teams per conference. Use U(21) to prove that a subgroup of order 7 cannot exist.
What is because ∣U(21)∣=12, so subgroup orders can only be 1, 2, 3, 4, 6, or 12?
The AFC East (4 teams) inside the AFC (16 teams) has index 4. Explain what this means in terms of cosets and NFL divisions.
What is that the AFC is partitioned into 4 cosets, just like the 4 divisions of the conference?
U(20) has 8 elements. Find one nontrivial subgroup and explain why its order divides 8.
What is, for example, {1,9,11,19}, a subgroup of order 4, consistent with Lagrange’s theorem?
Let H={1,31}⊆U(32). Compute cosets of HH and explain why they form a partition of U(32).
What is cosets dividing the group into equal-sized pieces, matching how divisions split conferences?