Mathematics · Advanced university
Groups, subgroups and element order · revision notes
An associative operation has an identity and an inverse for every element; commutativity is optional.
Key ideas and conditions
Group
An associative operation has an identity and an inverse for every element; commutativity is optional.
Subgroup
A nonempty subset closed under products and inverses inherits the group structure.
Element order
The least positive power returning the identity is the finite order; some elements have infinite order.
Lagrange
The order of a subgroup of a finite group divides the group order.
Explain without looking
- Why does Lagrange's theorem not say that every divisor is the order of a subgroup in every finite group?
- What is the order of the identity element?
Further reading and review references
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