4) Cosets and Index

Question 1

What is a coset

Answer 1

Question 2

Under what condition is a subgroup
H of a group G one of its own cosets

Answer 2

The subgroup H is always one of its own cosets, specifically the coset formed by multiplying
H by the group’s identity element (eH=H)

Question 3

When does a left coset of a subgroup
H in a group G equal the subgroup itself

Answer 3

gH = H if and only if g ∈ H

Question 4

When is an element g always included in its corresponding left coset gH

Answer 4

For all g ∈ G, we have g ∈ gH, because
g = ge where e ∈ H

Question 5

What condition determines when two left cosets of a subgroup H in a group G are equal

Answer 5

Question 6

Describe the proof of the coset equality condition

Answer 6

Question 7

What is the relationship between any two left cosets of a subgroup H in a group G

Answer 7

Let G be a group, H ≤ G, and x, y ∈ G.
Then either xH = yH or xH ∩ yH = ∅
(Cosets are equal or disjoint)

Question 8

Describe the proof that cosets are equal or disjoint

Answer 8

• If two left cosets xH and yH in a group G share at least one element, they are equal.
• This follows because the presence of a common element z in both xH and yH implies xH = zH and yH = zH using the coset equality condition. Therefore, xH = yH.
• If no common element exists, the cosets are disjoint

Question 9

How do the cosets of a subgroup
H partition the group G

Answer 9

If H ≤ G, then G is the disjoint union of the distinct left cosets of H in G

Question 10

What is the index of a subgroup

Answer 10

The cardinality of the set of left cosets of H in G and is denoted by [G : H]

Question 11

Do all cosets of a subgroup H in a group G have the same size

Answer 11

Yes, |gH| = |H| for all g ∈ G

Question 12

Describe the proof that the cosets of a subgroup H in a group G have the same cardinality

Answer 12

Question 13

What is Lagrange’s Theorem

Answer 13

Question 14

Describe the proof of Lagrange’s Theorem

Answer 14

• Start by partitioning G into r = [G:H] distinct, non-overlapping left cosets of H
• Since all cosets have the same number of elements as H, the total number of elements in G is the sum of the sizes of these cosets, which equals r times ∣H∣
• Thus, ∣G∣=[G:H]×∣H∣, demonstrating that both ∣H∣ and [G:H] divide ∣G∣

Question 15

In a finite group G, if g is an element of G, what relationship exists between g| and |G|

Answer 15

|g| divides |G|

Question 16

What property do G-th powers exhibit in a finite group G

Answer 16

Question 17

What is the exponent of a group

Answer 17

The term for the smallest positive integer k such that
g ^k =e for all elements g in a group G

Question 18

If the order of a group is prime what does that imply about the group

Answer 18

The group is cyclic

Question 19

Describe the proof that every group of prime order is cyclic

Answer 19