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The related but different uniform word problem for a class of recursively presented groups is the algorithmic problem of deciding, given as input a presentation for a group in the class and two words in the generators of , whether the words represent the same element of .
Given a presentation for a group G, the word problem is the algorithmic problem of deciding, given as input two words in S, whether they represent the same element of G. The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911.
One also says that the latter is a quotient group of the former, because some once different elements become equal in the new group. However, it is also a subgroup, because we can simply fill the missing component with 0 {\displaystyle 0} to get back to Z n ⊕ Z n {\displaystyle \mathbb {Z} _{n}\oplus \mathbb {Z} _{n}} .
Given a population of 13 coins in which it is known that 1 of the 13 is different (mass) from the rest, it is simple to determine which coin it is with a balance and 3 tests as follows: 1) Subdivide the coins in to 2 groups of 4 coins and a third group with the remaining 5 coins. 2) Test 1, Test the 2 groups of 4 coins against each other: a.
Every group has a presentation. To see this, given a group G, consider the free group F G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism.
Content delivery networks that distribute much of the world's content and services solve this large and complex stable marriage problem between users and servers every tens of seconds to enable billions of users to be matched up with their respective servers that can provide the requested web pages, videos, or other services.
The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements and of a group are equal, because the group axioms imply = = . It is thus customary to speak of the identity element of the group. [27]
In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups.. The isomorphism problem was formulated by Max Dehn, [1] and together with the word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911. [2]