An Interpretive Account: The Analysis

The Details of the Argument for GRF5.6 that Sparked the Controversy. As illustrated in fig.3, the argument for GRF5.6 (Prove that a group of order 35 contains elements of order 5 and of order 7) presented in the tutorial is as follows:

Jack's argument (J1) grasps the essential part of proving that, in all cases, G contains elements of order 5 and 7, but is not clear in his presentation of the cases he takes. However, as the tutor notes (T1 and T2), selecting an element of order 5 and proving that the rest of the elements in G are of order 5 or 7, by not excluding the possibility that among them there maybe an element of order 35, is not legitimate. In Jack's mind this exclusion seems to have taken place in the beginning, when he let xÎG and lateralised the cases for |x|. But this is different from what the tutor says: choosing an element in terms of its order says nothing about the order of the rest of the elements in G. Jack proves that, in case |x|=35, then G is cyclic; therefore it contains elements of order 5 (x⁷) and of order 7 (x⁵). When he chooses |x|=5, he does not exclude the possibility of another yÎG whose order is neither 5 nor 7 and g¹e.

In a sense, the presentation of the argument as in fig.3 or above is less controversial and clearer than Jack's. It seems that the exclusion process described above has taken place in Jack's mind (J2 can be taken as a declaration of his taking mutually exclusive cases and that once a case has been looked at, for instance |x|=35, it is not necessary that it reappears in the proof) but is not emphasised enough in his presentation.

Lateralising cases in terms of G being or not being cyclic, is, as the tutor's argument shows in T1 and T2, a more unambiguous approach. Jack's however lateralisation seems more natural: the question asks for a proof that a group of order 35 contains elements of order 5 and elements of order 7. So Jack sets out to point these elements out. This forces him to take cases. In the course of his checking out all the possibilities, if he were more precise, he would have to distinguish between G being or not a cyclic group. So the distinction, if we follow Jack's train of thought, is born out of the argument in his proof as a necessity; it is deeply incidental.

In contrast, the tutor's argument, as succinctly put in T2, suggests that the proof should start with making the distinction between the case where G is cyclic and G is not cyclic. This fine, logical suggestion however is the benefit of hindsight: knowing that subsequently the proof requires a lateralisation of cases, the tutor suggests that this can be neatly done in the beginning. There doesn't seem to be any psychological reason why someone would start this proof from distinguishing between cyclic and non-cyclic groups.

The above juxtaposition illustrates the contrast between expert and novice approaches: the former logically economical, succinct and benefiting from hindsight; the latter naturalistic and exploratory. Jack in the end doesn't appear as if he is clear about the difference between his approach and the tutor's. In fact the tutor, who calls this difference a 'minor point', is not emphasising the logical underpinnings of his and of Jack's approach: he seems to be more preoccupied with the full coverage of the cases in a formal way (he insists that Jack is clear that, when he deduces the existence of an element of order 7 from the existence of an element of order 5, he has excluded the possibility of the existence of an element of order 35) and less with the naturality in the genesis of the proving argument.