AS October 2020 Q1
1. The set \(G = \{1, 3, 7, 9, 11, 13, 17, 19\}\) under the binary operation of multiplication modulo 20 forms a group.
(a) Find the inverse of each element of \(G\). (3)
(b) Find the order of each element of \(G\). (3)
(c) Find a subgroup of \(G\) of order 4 (1)
(d) Explain how the subgroup you found in part (c) satisfies Lagrange’s theorem. (1)
| Scheme | Marks | AO | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1, 9, 11 and 19 are self-inverse | M1 A1 | 1.1b 1.1b | ||||||||
| B1 | 1.1b | ||||||||
| (3) |
Notes
M1: For any 2 of the self-inverse elements
A1: All 4 self-inverse elements correctly identified
B1: Correct inverses for the other elements
| Scheme | Marks | AO | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 A1 A1 | 1.1b 1.1b 1.1b | ||||||||||||||||
| (3) |
Notes
M1: At least 3 correct orders
A1: 6 correct orders
A1: All correct
| Scheme | Marks | AO |
|---|---|---|
| \(\{1, 3, 7, 9\}\) or \(\{1, 9, 13, 17\}\) or \(\{1, 9, 11, 19\}\) | B1 | 2.5 |
| (1) |
Notes
B1: Describes a correct subgroup of order 4
| Scheme | Marks | AO |
|---|---|---|
| Because 4 is a factor of 8 | B1 | 2.4 |
| (1) | ||
| (8 marks) |
Notes
B1: Correct explanation