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Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the identity of the group G, for each i(1 sis n). Then () For each i, H. = {(e,, e, ... , e X, e;+ 1., e, | x; eG)} is a normal subgroup of G

Let G, (Isisn) be n groups and G is the external direct product of these groups. Let e, be the identity of the group G, for each i(1 sis n). Then () For each i, H. = {(e,, e, ... , e X, e;+ 1., e, | x; eG)} is a normal subgroup of G

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter1: Systems Of Linear Equations
Section1.1: Introduction To Systems Of Linear Equations
Problem 89E: Show that if ax2+bx+c=0 for all x, then a=b=c=0.
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Let G, (1 <isn) be n groups and G is the external direct product of these groups. Let e, be the identity of the group G, for each i(1 sisn). Then () For each i, H,= {(e,, e, ., e*p e;+1, e, x; eG} is a normal subgroup of G .... (i) H, is isomorphic to G, i.e. H;= G,,i (ii) Each g e G can be written uniquely as product of elements from H, H, .. Hn.
Transcribed Image Text:Let G, (1 <isn) be n groups and G is the external direct product of these groups. Let e, be the identity of the group G, for each i(1 sisn). Then () For each i, H,= {(e,, e, ., e*p e;+1, e, x; eG} is a normal subgroup of G .... (i) H, is isomorphic to G, i.e. H;= G,,i (ii) Each g e G can be written uniquely as product of elements from H, H, .. Hn.
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