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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 conjecture asks whether a finite subgroup of the mapping class group of a surface can be realized as the isometry group of a hyperbolic metric on the surface. The mapping class group acts on Teichmüller space. An equivalent way of stating the question asks whether every finite subgroup of the mapping class group fixes some point of ...
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.
Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H. Its index in G must also correspond to a subgroup of the symmetric group S n, the group of permutations of n objects. So for example if n is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S 5.
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 maximal compact subgroup is a maximal subgroup amongst compact subgroups – a maximal (compact subgroup) – rather than being (alternate possible reading) a maximal subgroup that happens to be compact; which would probably be called a compact (maximal subgroup), but in any case is not the intended meaning (and in fact maximal proper subgroups are not in general compact).
A search problem is often characterized by: [1] A set of states; A start state; A goal state or goal test: a boolean function which tells us whether a given state is a goal state; A successor function: a mapping from a state to a set of new states