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We now look at some infinite subgroups of *I*(**R**^{2}).

We will look at subgroups which map a strip, say the subset **R** × [-1, 1], and are *discrete*. That is, there are no elements which move points an arbitrarily small distance.

[For example, the additive group of reals **R** acting on the line by addition is *not* a discrete group.]

The *finite* symmetry group of the strip is *D*_{2}. The elements of this group are: the identity *I*, a reflection *R* in the horizontal, a reflection *V* in a vertical line (through **0** say) and a half turn *H* (about **0** say).

We have *R*^{2} = *V*^{2} = *H*^{2} = *I* , *RV* = *VR* = *H* etc.

What else could be in an infinite group?

It could contain a translation *T* (which we will take to be by a shortest distance in the group) and/or a glide reflection *G* (which we can take to be *G* = *TR* = *RT*).

We then get the following possible groups generated by some of these elements. In each case we give a pattern which is mapped to itself by the group.

(See Exercises 4 Question 3)

Group generated by: | Pattern left invariant | Group | |

(i) | T | .... .... | C_{∞} |

(ii) | G | .... .... | C_{∞} |

(iii) | T, R | .... .... | C_{∞} × D_{1} |

(iv) | T, V | .... .... | D_{∞} |

(v) | T, H | .... .... | D_{∞} |

(vi) | G, V | .... .... | D_{∞} |

(vii) | everything | .... .... | D_{∞} × D_{1} |

These seven groups are the only possibilities -- up to conjugacy in the group

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