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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 .
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.,
An important example in the theory of Lie groups arises when is taken to be (;), the group of invertible matrices with complex entries. In that case, a basic result is the following: [ 5 ] Theorem : Suppose φ : R → G L ( n ; C ) {\displaystyle \varphi :\mathbb {R} \rightarrow \mathrm {GL} (n;\mathbb {C} )} is a one-parameter group.
The subgroup method is an algorithm used in the mathematical field of group theory. It is used to find the word of an element. It doesn't always return the minimal word, but it can return optimal words based on the series of subgroups that is used. The code looks like this:
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) .
For example, any subgroup of the group of integers (, +) is generated by some integer . If = then the subgroup takes up 0 proportion. Otherwise, it takes up / of the whole group. Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.
C 2 is the same as B 2, and C 1 is the same as B 1 and A 1. D n I (n ≥ 4) split n(2n - 1) n: SO(n)SO(n) Order 4 if n odd, 8 if n even 2 if n > 4, S 3 if n = 4: identity component of projective special orthogonal group PSO(n,n) n 2: D 3 is the same as A 3, D 2 is the same as A 1 2, and D 1 is abelian. E 6 6 I split 78 6 C 4: Order 2 Order 2 E ...
Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions. The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov 's notion of word growth .