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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 .
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 generated by u 1, ..., u n. [18] The group G is subgroup separable, that is, every finitely generated subgroup is closed in the pro-finite topology on G ...
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.
Subgroup analysis refers to repeating the analysis of a study within subgroups of subjects defined by a subgrouping variable. For example: ...
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 ...
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 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.
This is the problem of groups with a strongly p-embedded 2-local subgroup with p odd, which was handled by Aschbacher. Quasithin groups. A quasithin group is one whose 2-local subgroups have p-rank at most 2 for all odd primes p, and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher and Smith ...