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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 order-finding problem can also be viewed as a hidden subgroup problem. [3] To see this, consider the group of integers under addition, and for a given a ∈ Z {\displaystyle a\in \mathbb {Z} } such that: a r = 1 {\displaystyle a^{r}=1} , the function
Simon's problem considers access to a function : {,} {,}, as implemented by a black box or an oracle. This function is promised to be either a one-to-one function, or a two-to-one function; if is two-to-one, it is furthermore promised that two inputs and ′ evaluate to the same value if and only if and ′ differ in a fixed set of bits. I.e.,
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.
Additionally, a family of sets may be defined as a function from a set , known as the index set, to , in which case the sets of the family are indexed by members of . [1] In some contexts, a family of sets may be allowed to contain repeated copies of any given member, [ 2 ] [ 3 ] [ 4 ] and in other contexts it may form a proper class .
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.
In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set. In mathematical analysis and in probability theory , the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection.
For example, consider the infinite cyclic group ℤ = b , embedded as a normal subgroup of the Baumslag–Solitar group BS(1, 2) = a, b . With respect to the chosen generating sets, the element b 2 n = a n b a − n {\displaystyle b^{2^{n}}=a^{n}ba^{-n}} is distance 2 n from the origin in ℤ , but distance 2 n + 1 from the origin in BS(1, 2) .