Section (i) A Gradually Revealing Example of the Linguistic and Conceptual Dimensions of Difficulty With Order of an Element, Generating <g> and the Group Operation

An Interpretive Account: The Analysis

This episode stands as a metaphor for a cognitive journey back to the roots of the student's confusion with the new group theoretical concepts she has been introduced to: ostensibly about GRF5.1, this is mostly about <g>, |g| and the group operation.

A Problematic Use of the Term Order of an Element. As the tutor notes (his observation is based on Connie's reaction to Alan's solution in the group tutorial that morning) Connie lacks a clear understanding of the notion of order of an element. He then chooses to re-introduce her to the notion, via the definition given in the lectures, and explain that |g|=|<g>|. In C1 and C2 Connie seems to be slightly surprised with the connection between |g| and <g>.In fact her confusion and surprise can be attributed to the linguistic use of the term order of an element g which actually is an abbreviation of the more accurate order of the group generated by an element g. C4 illustrates how Connie is in trouble with understanding how an element can have an order — which so far has been identified as a property of groups: groups have order and the order of a group is a number equal to the number of its elements. Connie's confusion can be accounted for as an effect of her not realising the tacit abbreviating of the term. C3 is evidence of her seeing order as the number of elements in a group and actually as a finite number: she uses the word 'count' in order to talk about the act of finding the order of a group. I note that her finitist conception of order is perfectly justifiable in the context of finite groups she has been recently working in.

Finding however the order of a cyclic group by counting the number of its elements obscures the fact that if <g> is of order p that means that it has p elements because the powers of g start 'repeating themselves' after p. In this sense order of an element is a concept that contains both a static characteristic of <g> (the number of its elements) AND information about the process of obtaining these elements (how many times it is necessary to take the powers of g in order to cover <g>). In Connie's words, most strikingly C4, this duality is missing.

Generating <g> and the Notion of Group Operation. Subsequently in C5 to C9, it turns out that behind Connie's unease with the concept of order lies the even more fundamental unease with the notion of generating a cyclic group from the powers of an element gÎG. Even further her troubled notion of generating can be largely attributed to her muddled perception of the operation in a group.

In C5-C8 'times' and 'to the power of' are used unclearly interchangeably; it is also not clear at all in each one of them whether Connie refers to the 'multiples' or 'powers' of g or g². This can be due to the tutor's effort to convey the idea that, in a subgroup of order p, where p is a prime number, every element, other than e, is of order p too. Therefore, whether we consider the powers of g or g², eventually the generated set will be H. In this sense the tutor and Connie are far from understanding each other and communicating fruitfully: Connie is still struggling with her exploration of the notion of generating a group from an element; she is still vague about how this process takes place (operational stage). The tutor on the other hand assumes the clarity of the process of generating a group from g and a group from g² and attempts a demonstration of how these two groups coincide.

It seems fair to say that the discourse between the two interlocutors takes place past each other. Perhaps most striking is the exchange of words in C5 to C8 with regard to 'times' and 'to the [power of]': in C5 Connie explicitly uses 'times' and in T2 the tutor responds with taking powers. In C6 and C7 she insists on 'times' and the tutor shifts from 'multiplying' elements that are powers of g to manipulating the powers to which these elements are taken. While doing so he seems to assume the clarity of these operations. C9 is evidence of how Connie, far into the discussion, is still struggling with clarifying the objects on which the operations are applied. Since the conversation is completed with C10 — it seems that the tutor has been expecting a verbal signal of understanding from Connie so that he can move on to other topics — there is no evidence of whether Connie's perception of the group operation, the generating process and ultimately cyclic groups and their order has been clarified and enriched.