the second map corresponds to the projection given by the action of the

quotient group [Z.sub.2] = O(d)/SO(d) on the quotient space [S.sup.c.sub.n]([R.sup.d])/SO(d) (SO(d) is normal in O(d)).

Milnor's K-group is introduced in [2] as the

quotient group of the tensor product [k.sup.*] [cross product] ...

A counterexample is the

quotient group of the Heisenberg group by an infinite cyclic group (see [5, p.

Throughout this paper, a

quotient group G := G/K, where K [??] G [less than or equal to] Sym([OMEGA]), is specified by a pair consisting of generating sets for G and K, and an element g [member of] G is specified by a coset representative g [member of] G such that g = Kg.

Given a group G suppose it has a normal subgroup N [??] G, and let U and S be any groups such that U isomorphic with N and S is isomorphic with the

quotient group G/N.

The group of cosets {H+a: a [member of G} is called the

quotient group and is written G/H.

These three cosets form the so-called

quotient group [C.sub.12]/[C.sub.4].

Let F(k) be the free group in k generators [[alpha].sub.1], [[alpha].sub.2],..., [[alpha].sub.k], and RF(k) the

quotient group obtained from F(k) by adding relations which say that each [[alpha].sub.i] commutes with all of its conjugates.

We denote by [[pi].sup.c.s.sub.1] (X) the

quotient group of [[pi].sup.ab.sub.1] (X) which classifies abelian c.s coverings of X.

It introduces group actions early, and includes Hasse diagrams of posets and homomorphism diagrams, while introducing normal subgroups,

quotient groups, and homomorphisms late.

Thus far we have considered

quotient groups in terms of group tables, but there is an important theorem which is reflected in the subgroup diagram, namely the lattice isomorphism theorem.

In this study, we explore six students' conceptions of Z[subscript n] in an effort to understand students' conceptions of

quotient groups in general.