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

Describe the proof that gH = H if and only if g ∈ H

Answer 4

Question 5

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

Answer 5

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

Question 6

Find an example of a group G with a subgroup H such that left and
right cosets are different

Answer 6

Question 7

What is the coset equality condition

Answer 7

Question 8

Describe the proof of the coset equality condition

Answer 8

Question 9

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

Answer 9

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

Question 10

Describe the proof that cosets are equal or disjoint

Answer 10

• 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 11

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

Answer 11

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

Question 12

What is the index of a subgroup

Answer 12

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

Question 13

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

Answer 13

Yes, |gH| = |H| for all g ∈ G

Question 14

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

Answer 14

Question 15

What is Lagrange’s Theorem

Answer 15

Question 16

Describe the proof of Lagrange’s Theorem

Answer 16

• 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 17

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

Answer 17

|g| divides |G|

Question 18

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

Answer 18

Question 19

What is the exponent of a group

Answer 19

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

Question 20

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

Answer 20

The group is cyclic

Question 21

Describe the proof that every group of prime order is cyclic

Answer 21