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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.
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, 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) .
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.
Inside the 'Fast and Furious' Cast's Relationship With Meadow Walker. Read article. Brian’s life at home is still running smoothly in Fast X, which hit theaters on Friday, May 19.While there is ...
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R).The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces.