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The free group FS is defined to be the group of all reduced words in S, with concatenation of words (followed by reduction if necessary) as group operation. The identity is the empty word. A reduced word is called cyclically reduced if its first and last letter are not inverse to each other.
Create distribution lists to save time when you send emails to a group of contacts from the contacts you already have in your AOL Contacts, set up a contact list with a group of people you often send emails. For example, you email the same content to 3 friends every week. Instead, create a contact list called "Friends". Send one email to your ...
Create Music Group, formerly known as CreateTV, is an independent American music distribution, publishing, and data analytics company founded in 2015 by Jonathan Strauss, Alexandre Williams & Wayne Hampton. The company is headquartered in Los Angeles, California. The company provides music distribution, rights management and music publishing.
A normal form for a free product of groups is a representation or choice of a reduced sequence for each element in the free product . Normal Form Theorem for Free Product of Groups. Consider the free product of two groups and . Then the following two equivalent statements hold. (1) If , where is a reduced sequence, then in.
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. Then K is normal in F G, therefore is equal to its normal closure, so G | K = F G /K.
A presentation of a group by generators corresponds to a surjective homomorphism from the free group on generators to the group , defining a map from the Cayley tree to the Cayley graph of . Interpreting graphs topologically as one-dimensional simplicial complexes , the simply connected infinite tree is the universal cover of the Cayley graph ...
The Nielsen–Schreier theorem states that if H is a subgroup of a free group G, then H is itself isomorphic to a free group. That is, there exists a set S of elements which generate H, with no nontrivial relations among the elements of S . The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the ...
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses . In other words, if is a subset of a group , then , the subgroup generated by , is the smallest subgroup of ...