Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e, b)| be G}, where e, and ez are identity elements of G, and G, respectively, then H, and H, are normal subgroups of G, x (ii) H G, and H, G, (v) The factor (quotient) group (G, xG2)| H¡ is isomorphic to G2, and (G, x G2)| H, is isomorphic to G,.
Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e, b)| be G}, where e, and ez are identity elements of G, and G, respectively, then H, and H, are normal subgroups of G, x (ii) H G, and H, G, (v) The factor (quotient) group (G, xG2)| H¡ is isomorphic to G2, and (G, x G2)| H, is isomorphic to G,.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 14E: Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is...
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Transcribed Image Text:Theorem 2. Let G, and G, be groups, then
@ Gx G, G, x G,
(6) If H = {(4, e)| a e G} and H, = {(e, b)| be G, where e, and e, are identity
elements of G, and G, respectively, then H, and H, are normal subgroups of G x
(ii) H G, and H, G,
(iv) The factor (quotient) group (G, xG2)| H; is isomorphic to G2, and (G, × G) H, is
isomorphic to G,.
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