mathsfun Posted November 17, 2004 Share Posted November 17, 2004 can somebody help? prove or disprove, If G is a group in which every proper subgroup is cyclic, then G is cyclic. in additon, can u explain cyclic Link to comment Share on other sites More sharing options...
mathsfun Posted November 17, 2004 Author Share Posted November 17, 2004 can somebody help? prove or disprove, If G is a group in which every proper subgroup is cyclic, then G is cyclic. in additon, can u explain cyclic Link to comment Share on other sites More sharing options...
bloodhound Posted November 17, 2004 Share Posted November 17, 2004 What is a "proper" subgroup? never came across that term before. If G is a group, and you have a [math]g \in G[/math] such that [math]G=<g>[/math], then G is called a cyclic group and g is called the generator of G Here, [math]<g>=\{g^k:k \in \mathbb{Z}\}[/math] or additively [math]<g>=\{mg:m \in \mathbb{Z}\}[/math] Well basically its just saying that, if u have a group G, then its a cyclic group if there exists an element in G such that if u take all the integer powers of that element, you get the group G. A nice property is that a subgroup of a cyclic group is also cyclic. Cyclic groups are also abelian by the way. Link to comment Share on other sites More sharing options...
bloodhound Posted November 17, 2004 Share Posted November 17, 2004 What is a "proper" subgroup? never came across that term before. If G is a group, and you have a [math]g \in G[/math] such that [math]G=<g>[/math], then G is called a cyclic group and g is called the generator of G Here, [math]<g>=\{g^k:k \in \mathbb{Z}\}[/math] or additively [math]<g>=\{mg:m \in \mathbb{Z}\}[/math] Well basically its just saying that, if u have a group G, then its a cyclic group if there exists an element in G such that if u take all the integer powers of that element, you get the group G. A nice property is that a subgroup of a cyclic group is also cyclic. Cyclic groups are also abelian by the way. Link to comment Share on other sites More sharing options...
Dogtanian Posted November 17, 2004 Share Posted November 17, 2004 At a guess bloodhound, I say a "proper" subgroup was just a subgroup of G that in't G itself, saying that though, I don't think I've conme across the term before either...*thinks*... Link to comment Share on other sites More sharing options...
Dogtanian Posted November 17, 2004 Share Posted November 17, 2004 At a guess bloodhound, I say a "proper" subgroup was just a subgroup of G that in't G itself, saying that though, I don't think I've conme across the term before either...*thinks*... Link to comment Share on other sites More sharing options...
bloodhound Posted November 17, 2004 Share Posted November 17, 2004 oh rite... maybe its equivalent to a "proper" subset. cheers .. ill have a go at doing this question. Link to comment Share on other sites More sharing options...
bloodhound Posted November 17, 2004 Share Posted November 17, 2004 oh rite... maybe its equivalent to a "proper" subset. cheers .. ill have a go at doing this question. Link to comment Share on other sites More sharing options...
matt grime Posted November 17, 2004 Share Posted November 17, 2004 what's your inkling? do you think it true or false? Link to comment Share on other sites More sharing options...
matt grime Posted November 17, 2004 Share Posted November 17, 2004 what's your inkling? do you think it true or false? Link to comment Share on other sites More sharing options...
Woxor Posted November 30, 2004 Share Posted November 30, 2004 At a guess bloodhound, I say a "proper" subgroup was just a subgroup of G that in't G itself, saying that though, I don't think I've conme across the term before either...*thinks*... Yes, that's right. The only real conceptual difference between that and a proper subset is that the trivial subgroup is still a proper subgroup, whereas the null set isn't a proper subset. Link to comment Share on other sites More sharing options...
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