What is the order of [4] in the additive group Z/10Z?
Answer and explanation
5
The least positive k with 10 dividing 4k is 5; equivalently 10/gcd(4,10)=5.
Subjects
Abstract algebra
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Sample exercises
Prove the identity element of a group is unique.
Answer and explanation
If e and f are both identities, then ef=f because e is an identity and ef=e because f is an identity, so e=f.
Credit both identity properties.
Must every group be commutative?
- Only infinite groups are groups
- No
- Yes by associativity
- Yes if it has an identity
Answer and explanation
No
Commutativity is an additional property defining an abelian group.
A finite group has 18 elements. A subgroup has 6 elements. How many left cosets does it have?
Answer and explanation
3
The left cosets partition the group into equally sized sets of 6.
Show that the set of even integers is a subgroup of the additive integers.
Answer and explanation
It contains 0, sums of even integers are even, and additive inverses of even integers are even.
Credit identity and closure under addition and inverses.
Can a group of order 15 have a subgroup of order 4?
- Yes by Lagrange's theorem
- Always
- Only when abelian
- No
Answer and explanation
No
4 does not divide 15, so Lagrange rules it out.
Study notes
Groups, subgroups and element order
An associative operation has an identity and an inverse for every element; commutativity is optional.
Advanced universityGroups, subgroups and element order · revision notes
An associative operation has an identity and an inverse for every element; commutativity is optional.
Advanced universityHomomorphisms, kernels and quotient structure
A structure-preserving map respects the relevant operation.
Advanced universityHomomorphisms, kernels and quotient structure · revision notes
A structure-preserving map respects the relevant operation.
Advanced universityOrder of an element in an additive cyclic group
The operation is addition modulo n. Element order is the least positive multiple giving zero.
Advanced universityOrder of an element in an additive cyclic group · formula reference
The operation is addition modulo n. Element order is the least positive multiple giving zero.
Advanced universityDimension of a quotient vector space
W must be a linear subspace of a finite-dimensional vector space V.
Advanced universityDimension of a quotient vector space · formula reference
W must be a linear subspace of a finite-dimensional vector space V.
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