Lagrange Theorem and Counting Principles - One Line Questions
1.
If G is a group and H is a subgroup, and g is an element of G, what is a left coset of H in G? —
{h * g | h in H}
2.
Lagrange's Theorem also implies a relationship between the order of a group, the order of a subgroup, and the index of the subgroup. What is this relationship? —
|G| = |H| * [G:H]
3.
If G is a finite group and H is a subgroup of G, then Lagrange's Theorem states that: —
|H| divides |G|
4.
If a group G has order 10, which of the following cannot be the order of a subgroup of G? —
7
5.
What is the order of the element (1 2 3) in the symmetric group S_3? —
3
6.
Consider the group Z_6 under addition modulo 6. What is the order of the element 4? —
3
7.
What is the order of the group of integers modulo n under addition, Z_n? —
n
8.
Consider the group of units modulo 5, U(5) = {1, 2, 3, 4} under multiplication modulo 5. What is the order of the subgroup generated by 2? —
4
9.
How many non-isomorphic groups of order 4 exist? —
2
10.
What is the total number of ways to arrange the letters in the word 'MISSISSIPPI'? —
11! / (4! * 4! * 2! * 1!)
11.
What is the order of the alternating group A_4? —
12
12.
If H is a subgroup of G, and |G| = 24 and |H| = 6, what is the index [G:H]? —
4
13.
A password must be 4 characters long and can consist of uppercase letters or digits. How many possible passwords are there? —
(26+10)^4
14.
Consider the group of integers modulo 12 under addition, Z_12. What is the order of the subgroup generated by 3, which is {0, 3, 6, 9}? —
4
15.
Consider the symmetric group S_3. What is its order? —
6
16.
Consider the group of symmetries of an equilateral triangle (D_3). What is its order? —
6
17.
If there are 3 routes from city A to city B, and 4 routes from city B to city C, how many distinct ways are there to travel from city A to city C via city B? —
3 * 4 = 12
18.
In how many ways can the letters of the word 'MATH' be arranged? —
24
19.
How many ways can you arrange 5 distinct books on a shelf? —
120
20.
A committee of 3 people is to be selected from a group of 5 people. How many different committees can be formed? —
5 choose 3 (C(5,3))
21.
Consider forming a 3-digit number using digits 1, 2, 3, 4, 5 without repetition. How many such numbers can be formed? —
5 * 4 * 3
22.
How many distinct permutations are there of the letters in the word 'APPLE'? —
5! / 2!
23.
If G is a group of order 12, which of the following is a possible order for an element in G? —
4
24.
If there are 7 choices for the first letter of a word and 5 choices for the second letter, how many 2-letter words can be formed if repetition is allowed? —
7 * 5
25.
How many binary strings of length 8 are there? —
2^8
26.
The set of all permutations of n distinct objects forms which algebraic structure? —
A non-abelian group
27.
Which of the following is a subgroup of S_3? —
A cyclic group of order 3
28.
Which counting principle is used when calculating the number of possible outcomes when rolling two distinct dice? —
Multiplication Principle
29.
If a group G has order 30, what are the possible orders of its subgroups? —
Any divisor of 30
30.
In how many ways can a president, vice-president, and treasurer be selected from a club of 10 members? —
P(10,3)
31.
What is the number of permutations of n distinct objects taken k at a time? —
P(n,k)
32.
Which theorem is a generalization of Lagrange's Theorem for infinite groups? —
Cauchy's Theorem
33.
What is a necessary condition for a subgroup H to be a normal subgroup of G, related to cosets? —
Every left coset is equal to the corresponding right coset.
34.
Which of the following is NOT necessarily true for any finite group G? —
G has an element of order |G|.
35.
If a subgroup H has index 2 in G, what can be said about H? —
H is always a normal subgroup of G.
36.
What is the converse of Lagrange's Theorem? —
If a number 'n' divides the order of a group G, then G has a subgroup of order 'n'.
37.
What does the addition principle (or addition rule) state? —
38.
What is the fundamental counting principle (also known as the multiplication principle)? —
If an event can occur in 'm' ways and another independent event can occur in 'n' ways, then the two events can occur in sequence in m*n ways.
39.
What does Cauchy's Theorem state for finite groups? —
If G is a finite group and p is a prime dividing |G|, then G has an element of order p.
40.
What is the primary implication of Lagrange's Theorem for the structure of finite groups? —
It establishes that the order of every element and subgroup must divide the order of the group.
41.
If there are 'm' ways to do one thing and 'n' ways to do another, and these choices are independent, how many ways are there to do both? —
m * n
42.
If a set has n elements, how many subsets does it have? —
2^n
43.
What is the formula for combinations, 'n choose k' (C(n,k))? —
n! / (k! * (n-k)!)
44.
What is the formula for permutations, 'n permute k' (P(n,k))? —
n! / (n-k)!
45.
Lagrange's Theorem is a fundamental result in which area of mathematics? —
Abstract Algebra
46.
If you have 5 different flavors of ice cream and you want to choose 2 different flavors for a cone, how many combinations are possible? —
C(5,2)
47.
If G is a group of prime order p, then G is isomorphic to which group? —
The cyclic group Z_p
48.
In the context of Lagrange's Theorem, what is the relationship between the order of an element 'a' in a group G and the order of the group G? —
The order of 'a' divides the order of G.
49.
What is the statement of Lagrange's Theorem regarding the order of a subgroup? —
The order of a subgroup divides the order of the group.
50.
What is the index of a subgroup H in a group G, denoted by [G:H]? —
The number of distinct left (or right) cosets of H in G