Group theory MOC

Group order

The order |𝐺| of a group 𝐺 is the number of elements in that group. group Similarly order |𝑎| of an element 𝑎 ∈𝐺 is the smallest integer 𝑛 such that 𝑎𝑛 =𝑒, group where 𝑎 is said to have infinite order if no such 𝑛 exists. The reason for this dual naming and notation is the order of a cyclic group equals the order of its generator.

Properties

• The order of a subgroup divides the order of a group
• The order of an element and its inverse are the same
• 𝑎𝑏 and 𝑏𝑎 have the same order
• Elements of order greater than two come in pairs (𝑎,𝑎−1)
• The product of two finite-order elements may be infinite-ordered¹
• Number of elements of order 𝑑 in a finite group
• Order of powers of a group element
• Relationship between I𝑎𝑏I and I𝑎I𝑏I
• Cauchy’s order theorem

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tidy | en | SemBr

Footnotes

1. See Gallian §3 exercise 50 ↩