4) Cosets and Index Flashcards

1
Q

What is a coset

A
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2
Q

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

A

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)

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3
Q

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

A

gH = H if and only if g ∈ H

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4
Q

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

A
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5
Q

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

A

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

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6
Q

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

A
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7
Q

What is the coset equality condition

A
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8
Q

Describe the proof of the coset equality condition

A
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9
Q

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

A

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

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10
Q

Describe the proof that cosets are equal or disjoint

A
  • 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
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11
Q

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

A

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

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12
Q

What is the index of a subgroup

A

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

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13
Q

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

A

Yes, |gH| = |H| for all g ∈ G

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14
Q

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

A
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15
Q

What is Lagrange’s Theorem

A
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16
Q

Describe the proof of Lagrange’s Theorem

A
  • 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∣
17
Q

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

A

|g| divides |G|

18
Q

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

A
19
Q

What is the exponent of a group

A

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

20
Q

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

A

The group is cyclic

21
Q

Describe the proof that every group of prime order is cyclic

A