Groups are the building blocks of abstract algebra. Subgroups are smaller groups within larger ones, sharing the same operation. They must include the , be closed under the operation, and contain inverses for all elements.
Cyclic groups are special groups generated by a single element. Every element in a can be expressed as a power of the generator. Cyclic groups are always abelian and can be finite or infinite in order.
Subgroups and their properties
Definition and requirements
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A is a subset of a group that is itself a group under the same operation
The identity element of the group must be in the subgroup (e∈H)
If a and b are in the subgroup, then ab must also be in the subgroup (, a,b∈H⟹ab∈H)
If a is in the subgroup, then a−1 must also be in the subgroup (, a∈H⟹a−1∈H)
The holds for elements in the subgroup ((ab)c=a(bc) for all a,b,c∈H)
Examples of subgroups
The set of even integers under addition is a subgroup of the integers under addition (2[Z](https://www.fiveableKeyTerm:z)≤Z)
The set of matrices with determinant 1, known as the special linear group SL(n,R), is a subgroup of the general linear group GL(n,R)
Subgroup identification
Steps to determine if a subset is a subgroup
Check if the identity element of the group is in the subset (e∈H)
Verify that the subset is closed under the group operation (if a and b are in the subset, then ab must also be in the subset)
Ensure that for each element in the subset, its inverse is also in the subset (a∈H⟹a−1∈H)
If all three conditions are met, the subset is a subgroup; otherwise, it is not
Examples of subgroup identification
The set {1,−1,i,−i} under multiplication is a subgroup of the complex numbers under multiplication
The set of 2x2 matrices with determinant 0 is not a subgroup of GL(2,R) because it does not contain the identity matrix
Cyclic groups and generators
Definition and properties
A cyclic group is a group that can be generated by a single element, called a generator
If a is a generator of a cyclic group, then every element of the group can be written as a power of a (an for some integer n)
The order of a cyclic group is the smallest positive integer n such that an=e, where e is the identity element
A cyclic group is always abelian (commutative, ab=ba for all a,b in the group)
A group of prime order is always cyclic
A group of infinite order can be cyclic (the integers under addition, (Z,+))
Examples of cyclic groups
The group of integers under addition (Z,+) is cyclic, with generators 1 and -1
The group of complex nth roots of unity under multiplication is cyclic, with generator e2πi/n
Identifying cyclic groups and generators
Steps to prove a group is cyclic
To prove a group is cyclic, find an element that generates all other elements in the group
Start by computing powers of each element in the group until all elements are generated or a repetition occurs
If an element generates all other elements, the group is cyclic, and that element is a generator
If no single element generates the entire group, the group is not cyclic
Finding generators in a cyclic group
In a cyclic group of order n, the generators are the elements a such that gcd(∣a∣,n)=1, where ∣a∣ is the order of the element a
For example, in the cyclic group Z6 under addition, the generators are 1 and 5 because gcd(1,6)=1 and gcd(5,6)=1