Norm (group)
In group theory, a norm refers to a function that assigns a non-negative real number to each element of a group, providing a measure of the "size" or "length" of the element. This concept draws an analogy to norms in vector spaces. However, unlike vector space norms, there is no single, universally accepted definition of a norm for groups. Different definitions exist, tailored to specific contexts and applications.
One common approach involves defining a norm based on the word length of an element with respect to a given generating set. Let G be a group, and let S be a generating set for G. The word length, or norm, of an element g in G with respect to S, denoted as ||g||S, is the smallest non-negative integer n such that g can be expressed as a product of n elements from S ∪ S-1 (where S-1 is the set of inverses of elements in S).
This definition depends on the choice of the generating set S. Different generating sets may lead to different norms for the same element. A group is said to be finitely generated if it has a finite generating set. For finitely generated groups, the word length norm provides valuable information about the growth of the group.
Another, less common, usage of "norm" in the context of groups relates to the normal core of a subgroup. The normal core of a subgroup H of a group G is the intersection of all conjugates of H in G. It is the largest normal subgroup of G contained in H. This construction highlights the "normal" part contained within the subgroup, and, in some older literature, this normal core might be loosely referred to using the word "norm" in a specific historical context. It is crucial to differentiate this historical usage from the more modern definition based on word length with respect to a generating set.
The concept of norms in groups plays a crucial role in areas such as geometric group theory, combinatorial group theory, and the study of growth functions. The specific definition of the norm and its properties are critical to the analysis being performed.