Subgroups (AQA A-Level Further Maths): Flashcards
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10 cards from this deck
Define a subgroup
Define a subgroup
Subset of a group that forms a group under same operation
Trivial subgroup
Trivial subgroup
Subgroup containing only the identity element
Proper subgroup
Proper subgroup
Any subgroup that is not the entire group itself
Four conditions to verify subgroups (acronym)
Four conditions to verify subgroups (acronym)
NICE: Non-empty, Identity, Closure, Inverses
Counter-examples needed to disprove a subgroup
Counter-examples needed to disprove a subgroup
One (a single counter-example is sufficient)
Lagrange's theorem
Lagrange's theorem
Order of every subgroup divides the order of the group
Order of a group
Order of a group
The number of elements it contains
Element period/order property (Lagrange)
Element period/order property (Lagrange)
Period of any element is a factor of the order of
Identity element in subgroups
Identity element in subgroups
All subgroups must contain the identity element
For order-2 subgroup, property of non-identity element
For order-2 subgroup, property of non-identity element
Must be self-inverse