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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 .
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 .
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.
For a finite subgroup H of a finite group G, the index of H in G is equal to the quotient of the orders of G and H. isomorphism Given two groups (G, •) and (H, ·), an isomorphism between G and H is a bijective homomorphism from G to H, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given ...
Small groups of prime power order p n are given as follows: Order p: The only group is cyclic. Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p.
The group G has the congruence subgroup property: a subgroup H has finite index in G if and only if there is a positive integer n such that (). The group G has solvable subgroup membership problem , that is, there is an algorithm that, given arbitrary words w , u 1 , ..., u n decides whether or not w represents an element of the subgroup ...
Subgroup analysis refers to repeating the analysis of a study within subgroups of subjects defined by a subgrouping variable. For example: ...