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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 ...
Save yourself time when sending the same email to multiple people by creating a group of your contacts. Instead of adding each email address separately, you can email a bunch of contacts by typing your group's name in the "To" field of a new email. Once you've created a group, you can continue to add, edit, or delete contacts from it.
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 ...
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.
Presentation of a group. In mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators —so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation.
In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G, [1] or even in every group. [2] Words play an important role in the theory of ...
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