Group theory MOC

Coset

Given a subgroup 𝐻 ≤𝐺, the left coset of 𝐻 in an element 𝑔 ∈𝐺 is defined as group

𝑔𝐻={𝑔ℎ:ℎ∈𝐻}⊆𝐺

likewise the right coset as

𝐻𝑔={ℎ𝑔:ℎ∈𝐻}⊆𝐺

Properties

1. 𝑔 ∈𝐻 iff 𝑔𝐻 =𝐻𝑔 =𝐻 ( ⟹ by Reärrangement lemma, ⟸ by 𝑔𝑒 =𝑔)
2. As a consequence of the Reärrangement lemma |𝑔𝐻| =|𝐻|.
3. Cosets are either identical or disjoint.
4. Every 𝑔 ∈𝐺 is contained in at least one coset of 𝐻, namely 𝑔𝐻 (by 𝑔𝑒 =𝑔)
5. From 2–4, 𝐺 may be partitioned into equally sized cosets. Hence the order of a subgroup divides the order of a group.

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