Prasant36 Posted December 19, 2018 Share Posted December 19, 2018 Let G be a group in which, (ab)^2 = (a^2)(b^2) for all a,b ∈G . Show that H = { g^2|g ∈G } is a normal subgroup of G. Link to comment Share on other sites More sharing options...
taeto Posted December 20, 2018 Share Posted December 20, 2018 Where are you stuck, showing that H is a subgroup or showing that H is normal? Link to comment Share on other sites More sharing options...
Prasant36 Posted December 20, 2018 Author Share Posted December 20, 2018 Actually both. I can prove it by using axioms & definitions, 1st by showing that H is a sgp & then its normal. But the question holds only 2 marks. So I think there is a shorter way of doing it as well. I want help regarding that. Thank you. Link to comment Share on other sites More sharing options...
taeto Posted December 20, 2018 Share Posted December 20, 2018 I suspect you are supposed to see that G is abelian. That certainly helps for showing normality. 1 Link to comment Share on other sites More sharing options...
Prasant36 Posted December 20, 2018 Author Share Posted December 20, 2018 Okay. Thank you Sir. Link to comment Share on other sites More sharing options...
Martin Rattigan Posted July 29, 2020 Share Posted July 29, 2020 (edited) Deleted Edited July 29, 2020 by Martin Rattigan was rubbish Link to comment Share on other sites More sharing options...
joigus Posted July 29, 2020 Share Posted July 29, 2020 (edited) On 12/20/2018 at 9:10 AM, taeto said: I suspect you are supposed to see that G is abelian. That certainly helps for showing normality. Good tip.+1. Is this homework, @Prasant36? Edit: You also need Abelian character for showing closure Edited July 29, 2020 by joigus Addition Link to comment Share on other sites More sharing options...
ahmet Posted July 29, 2020 Share Posted July 29, 2020 (edited) 16 hours ago, joigus said: Edit: You also need Abelian character for showing closure in mathematics, closure can correspond many things.may I ask: which type of closure do you meantion here? Edited July 29, 2020 by ahmet Link to comment Share on other sites More sharing options...
joigus Posted July 29, 2020 Share Posted July 29, 2020 9 minutes ago, ahmet said: in mathematics, closure can correspond many things.may I ask: which type of closure do you meantion here? Take a guess. Link to comment Share on other sites More sharing options...
ahmet Posted July 29, 2020 Share Posted July 29, 2020 21 minutes ago, joigus said: Take a guess. I can guess many things really such as Algebraic closure, closure in topology and analysis , and functional analysis... Link to comment Share on other sites More sharing options...
joigus Posted July 29, 2020 Share Posted July 29, 2020 1 minute ago, ahmet said: Algebraic closure, Bingo!! "Is a group" refers to algebraic properties. \[g_{1},g_{2}\in H\Rightarrow g_{1}g_{2}\in H\] We're not talking topological groups. (I'm not aware that anybody mentioned a basis of neighbourhoods). Welcome to page 1. Neither have I read anything about a metric space. Link to comment Share on other sites More sharing options...
ahmet Posted July 29, 2020 Share Posted July 29, 2020 I think almost all parts of mathematics have intersections (even topology and functional analysis with algebra) .... Link to comment Share on other sites More sharing options...
joigus Posted July 29, 2020 Share Posted July 29, 2020 1 hour ago, ahmet said: in mathematics, closure can correspond many things.may I ask: which type of closure do you meantion here? Also, by "normal" (in this context) I understand: \[gHg^{-1}\subseteq H\] Not "perpendicular". Any more questions? Link to comment Share on other sites More sharing options...
ahmet Posted July 29, 2020 Share Posted July 29, 2020 Just now, joigus said: Also, by "normal" (in this context) I understand: gHg−1⊆H Not "perpendicular". Any more questions? no (more) questions ,I just tried to understand what you meant Link to comment Share on other sites More sharing options...
joigus Posted July 29, 2020 Share Posted July 29, 2020 Just now, ahmet said: I think almost all parts of mathematics have intersections (even topology and functional analysis with algebra) .... They do. I know, and you know. And I know you know. And you know I know you know. Can we stick to the topic, please? It's algebra. Group theory. That's why we are @ Linear Algebra and Group Theory normal subgroup problem 1 minute ago, ahmet said: no (more) questions ,I just tried to understand what you meant Ah, OK. I'm sorry if I misunderstood your question in any sense. Link to comment Share on other sites More sharing options...
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