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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: ...
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.
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 .
Larger samples may be an economic necessity or may be necessary to increase the area of opportunity in order to track very low nonconformity levels. [1] Examples of processes suitable for monitoring with a u-chart include: Monitoring the number of nonconformities per lot of raw material received where the lot size varies
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 .
A core-free subgroup is a subgroup whose normal core is the trivial subgroup. Equivalently, it is a subgroup that occurs as the isotropy subgroup of a transitive, faithful group action. The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
The simplification in a word problem induced by subgroup distortion suffices to construct a cryptosystem, algorithms for encoding and decoding secret messages. [4] Formally, the plaintext message is any object (such as text, images, or numbers) that can be encoded as a number n. The transmitter then encodes n as an element g ∈ H with word ...
The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory. One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H] is divisible by p, then the normalizer N = N G (H) of H in G is also such that [N : H] is divisible by p.