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Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order. [6]: 32 The concepts of abelian group and -module agree.
An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety.
Containment occurs exactly when S is abelian. If H is a subgroup of G, then N G (H) contains H. 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 ...
A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. [15] However, a characteristic subgroup of a normal subgroup is normal. [16] A group in which normality is transitive is called a T ...
In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group. Given a group G {\displaystyle G} , the center of G {\displaystyle G} , denoted as Z ( G ) {\displaystyle Z(G)} , is defined as the set of those elements of the group which commute with every element of the group.
normal subgroup A subgroup N of a group G is normal in G (denoted N G) if the conjugation of an element n of N by an element g of G is always in N, that is, if for all g ∈ G and n ∈ N, gng −1 ∈ N. A normal subgroup N of a group G can be used to construct the quotient group G / N. normalizer
Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite p-group is central and has order p. Indeed, the socle of a finite p-group is the subgroup of the center consisting of the central elements of order p. If G is a p-group, then so is G/Z, and so it too has a non-trivial center.
The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, / is abelian if and only if contains the commutator subgroup of . So in some sense it provides a measure of how far the group is from being abelian; the larger the ...