We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 05 June 2012
A group F is free with free generating set X if it possesses the following universal property: each function α: X → H of X into a group H extends uniquely to a homomorphism of F into H. We find in section 28 that for each cardinal C there exists (up to isomorphism) a unique free group F with free generating set of cardinality C. Less precisely: F is the largest group generated by X.
If W is a set of words in the alphabet X ∪ X-1, it develops that there is also a largest group G generated by X with w = 1 in G for each w ∈ W. This is the group Grp(X : W) generated by X subject to the relations w = 1 for w ∈ W.
In section 29 we investigate Grp(X: W) when X = {x1,…, xn] is finite and W consists of the words (xjxj)mij, for suitable integral matrices (mij). Such a group is called a Coxeter group. For example finite symmetric groups are Coxeter groups. We find that Coxeter groups admit a representation π: G → O(V, Q) where (V, Q) is an orthogonal space over the reals and Xπ consists of reflections. If G is finite (V, Q) turns out to be Euclidean space. Finite Coxeter groups are investigated via this representation in section 30, which develops the elementary theory of root systems.
The theory of Coxeter groups will be used extensively in chapter 14 to study the classical groups from a geometric point of view.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
