Pure Mathematics B4: Lagrange's Theorem

7 flashcards · Mathematics · by lakesare

1. Name 2 things you know about Lagrange. Joseph-Louis Lagrange (1736 - 1813, Italy)

   He came up with Lagrange's notation f'(x) for derivatives As opposed to Leibniz's notation dy/dx.He came up with Lagrange's theorem in group theory

2. [Lagrange's theorem] What's interesting about the orders of subgroups of the group S(□). S(□) contains 8 symmetries.

   Order of the subgroup always divides the order of the group. Joseph-Louis Lagrange (1736 - 1813, Italy)

3. [Lagrange's theorem] State Lagrange's theorem in own words.

   The size of the subgroup always divides the size of the group.

4. Why do they say "abstract groups" here?

   Abstract groups - (G, ◦). Specific/Concrete groups - S(□), S4.

5. [Lagrange's theorem] What's curious about the orders of subgroups of A4. A4 has order 12.

   There is no subgroup of order 6.

6. [Lagrange's theorem] Imagine visually why |subgroup| always divides |group| (Lagrange's Theorem).

   Because we can always build this rectangle.

7. Cayley's Theorem VS Lagrange's Theorem?

   Cayley's Theorem - every group ≅ permutation group It makes perfect sense that Cayley would think of a Cayley's theorem - it's seen in his Cayley's tables!Lagrange's Theorem - order of subgroup | order of G