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what is the difference between field and group in algebra

Please list three differences between "Group" and "Field". It really confuse me.

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marked as duplicate by Zev Chonoles Sep 2 '11 at 20:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

If you took the problem verbatim from a problem sheet, then my -1 for it as well. :-) I don't think it is a constructive question, since its not clear to me at all what would count as a valid answer and what would not. –  Srivatsan Sep 2 '11 at 19:57
A group can have $18$ elements. A field can't. A group can have $20$ elements. A field can't. A group can have $22$ elements. A field can't. And so on for $24$, $26$, $28$, $30$, not $32$, but yes for $34$, $36$, $38$. Ten differences already. –  André Nicolas Sep 2 '11 at 19:57
Not a good question, as it doesn't help to understand why either a Group or a Field might be a useful mathematical idea. It has a feel of accountancy rather than mathematics ... –  Mark Bennet Sep 2 '11 at 20:01
@André Nicolas:You reminded me of: the "Group" starts with 'G' "Field" not,The "Group" ends with 'p' field not ... as the group is mentioned within quotes I am treating it as asci strings ... –  Quixotic Sep 2 '11 at 21:06
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1 Answer

Under a formulation like that of Wikipedia

  1. A group has only one binary operation. A field has a total of four operations.

  2. A group has only one set for the operations. A field has two.

  3. A group only has three axioms. A field has many more.

As Bill correctly points out, though, the truth of each of these depends on the formulation used.

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The truth of the above statements depends on the set of axioms one employs. All of your statements may be false for certain sets of axioms. –  Bill Dubuque Sep 2 '11 at 20:03
Two sets for the operations? –  André Nicolas Sep 2 '11 at 20:05
@Andre If we have four operations for a field, then multiplication, addition, and subtraction all happen on one set. Division happens on another set. As Bill correctly points out the truth of what I wrote depends on the axiom set one employs. –  Doug Spoonwood Sep 2 '11 at 21:09
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