courses › Beyond College

Groups: orders, generators, and Lagrange’s theorem.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 58.

Sign in to start

Builds on: Congruences (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Order of an element par 20 s not tried yet

  worked example

  What is the order of 8 in the group ℤ₃₀ under addition mod 30?

  Answer: 15

  1. The order of k in ℤₙ is n ÷ gcd(k, n): the number of copies of k it takes to reach a multiple of n.
  2. gcd(8, 30) = 2, so the order is 30 ÷ 2 = 15.

• Generators par 20 s not tried yet

  worked example

  How many generators does the cyclic group ℤ₂₇ (addition mod 27) have?

  Answer: 18

  1. k generates ℤ₂₇ exactly when gcd(k, 27) = 1, so count those: φ(27).
  2. The only prime dividing 27 is 3, so φ(27) = 27 × (1 − 1/3) = 18.

• Order of a permutation par 25 s not tried yet

  worked example

  What is the order of the permutation (1 2 3 4 5)(6 7 8 9 10) in S₁₂?

  Answer: 5

  1. The cycles are disjoint, with lengths 5 and 5.
  2. Each cycle returns to its start after its own length, so all of them do together after lcm(5, 5) = 5.

• Possible subgroup sizes par 25 s not tried yet

  worked example

  A group has order 42. Which of these does Lagrange’s theorem rule out as the order of a subgroup?

  1. 3
  2. 7
  3. 15
  4. 14

  Answer: 15

  1. By Lagrange’s theorem, a subgroup’s order divides the group’s order.
  2. 3, 7 and 14 divide 42; 15 doesn’t (42 ÷ 15 leaves remainder 12).

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

No rounds yet.