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Subgroup analysis refers to repeating the analysis of a study within subgroups of subjects defined by a subgrouping variable. For example: smoking status defining two subgroups: smokers and non-smokers.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
For example, the subgroup Z 7 of the non-abelian group of order 21 is normal (see List of small non-abelian groups and Frobenius group#Examples). An alternative proof of the result that a subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in .
Graphic breakdown of stratified random sampling. In statistics, stratified randomization is a method of sampling which first stratifies the whole study population into subgroups with same attributes or characteristics, known as strata, then followed by simple random sampling from the stratified groups, where each element within the same subgroup are selected unbiasedly during any stage of the ...
Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions. The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov 's notion of word growth .
The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems such as factoring , discrete logarithm , graph isomorphism , and the shortest vector problem .
Each group is named by Small Groups library as G o i, where o is the order of the group, and i is the index used to label the group within that order.. Common group names: Z n: the cyclic group of order n (the notation C n is also used; it is isomorphic to the additive group of Z/nZ)
For a prime number, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group is a maximal-subgroup of , i.e., a subgroup of that is a p-group (meaning its cardinality is a power of ; or equivalently: For each group element, its order is some power of ) that is not a proper subgroup of any other -subgroup of .