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In other words, if is a subset of a group , then , the subgroup generated by , is the smallest subgroup of containing every element of , which is equal to the intersection over all subgroups containing the elements of ; equivalently, is the subgroup of all elements of that can be expressed as the finite product of elements in and their inverses ...
If S is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by S and is called the subgroup generated by S. An element of G is in S if and only if it is a finite product of elements of S and their inverses, possibly repeated. [6]
In this case, ST is the group generated by S and T; i.e., ST = TS = S ∪ T . If either S or T is normal then the condition ST = TS is satisfied and the product is a subgroup. [4] [5] If both S and T are normal, then the product is normal as well. [4] If S and T are finite subgroups of a group G, then ST is a subset of G of size |ST| given by ...
If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N G (H). If S is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S is the subgroup C G (S). A subgroup H of a group G is called a self-normalizing subgroup of G if N G (H) = H.
Given any subset of a group , the subgroup generated by consists of all products of elements of and their inverses. It is the smallest subgroup of G {\displaystyle G} containing S {\displaystyle S} . [ 33 ]
Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the ...
Given any set F of generators {}, the free group generated by F surjects onto the group G. The kernel of this map is called the subgroup of relations, generated by some subset D . The presentation is usually denoted by F ∣ D . {\displaystyle \langle F\mid D\rangle .}
For any subgroup of , the following conditions are equivalent to being a normal subgroup of .Therefore, any one of them may be taken as the definition. The image of conjugation of by any element of is a subset of , [4] i.e., for all .