Some Questions Related to
Lagrange’s Theorem (10/30)
• Let’s analyze A5 a bit:
• How many proper non-trivial divisors does |A5| have?
• A. 7
B. 8
C. 9
D. 10 E. 12
• How many different cycle structures exist in A5?
• A. 2
B. 3
C. 4
D. 5
E. 6
• How many 5-cycles are there in A5?
• A. 5
B. 10 C. 15 D. 24 E. 30
• How many 3-cycles are there in A5?
• A. 3
B. 6
C. 10 D. 15 E. 20
• How many pairs of 2-cycles are there in A5?
• A. 3
B. 6
C. 10 D. 15 E. 20
Questions continued
• How many subgroups of order 2 are there in A5?
• A. 0
B. 3
C. 10 D. 12 E. 15
• How many subgroups of order 3 are there in A5?
• A. 0
B. 10 C. 15 D. 20 E. 24
• How many cyclic subgroups of order 4 are there in A5?
• A. 0
B. 2
C. 4
D. 10 E. 12
• How many Klein 4-groups are there in A5?
• A. 0
B. 4
C. 5
D. 6
E. 8
• How many subgroups of order 5 are there in A5?
• A. 0
B. 3
C. 6
D. 12 E. 24
• How many subgroups of order 6 are there in A5?
• A. 0
B. 6
C. 10 D. 12 E. 15
And more...
• Does A5 have at least one subgroup of order 10?
• A. Yes
B. No (If yes, what is it isomorphic to?)
• Does A5 have at least one subgroup of order 12?
• A. Yes
B. No (If yes, what is it isomorphic to?)
• Does A5 have at least one subgroup of order 15?
• A. Yes
B. No
• Does A5 have at least one subgroup of order 20?
• A. Yes
B. No
• Does A5 have at least one subgroup of order 30?
• A. Yes
B. No
More Lagrange examples
• Suppose a group G has subgroup H and K with |H| = 15
and |K| = 8. What is |H
• A. 0
B. 1
C. 4
K| ?
D. 5
E. all are possible
• Suppose a group G has subgroup H and K with |H| = 20
and |K| = 10. What is |H
• A. 1
B. 2
C. 5
K| ?
D. 10
E. all are possible
• Suppose a group G of order 24 has subgroup H and K
with H
|K| ?
• A. 2
K
G and |H| = 4. Which is a possible order of
B. 6
C. 12
D. 16
E. all are possible
Fermat’s Little Theorem, Etc.
• What is 314 mod 13?
• A. 1
B. 3
C. 9
D. 12
E. 14
• If p is prime and 0 < a < p, what is a5p – 5 mod p ?
• A. 0
B. 1
C. a
D. ap E. Can’t be determined
• What is the order of U(24)?
• A. 4
B. 6
C. 8
D. 12
E. 24
• What is 716 mod 24?
• A. 1
B. 7
C. 10
D. 14
E. 21
• What is 7125 mod 24?
• A. 1
B. 7
C. 10
D. 14
E. 21
• These last three involve the “Euler-Fermat Theorem”:
If GCD(a, n) = 1, then a
(n) mod
n = 1.
Assignment for Friday
• Make sure all exercises in Chapter 7 already assigned are
done (assigned on Oct 28).
• Read the statement and proof of the “Groups of Order 2p”
Theorem (Theorem 7.3).