Group extension
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In mathematics, for G a group, G′ is an extension of G if there is an exact sequence
This situation is sometimes described by saying that G′ is an extension of G by H (though some authors, e.g., Weibel, define the exact sequence as the extension of H by G).
In other words: G' is a group, H is a normal subgroup of G' and the quotient group G'/H is isomorphic to group G. In contexts where the extension nomenclature is used, G and H are known and the properties of G' are to be determined.
One extension, the direct product, is immediately obvious. Several other general classes of extensions are known but no theory exists which treats all the possible extensions at one time. Group extension is usually described as a "hard" problem; it is termed the extension problem.
