Gr (?) Bner-shirshov Base Theory In The Group On A Number Of Applications

This thesis mainly introduce some applications of the Grobner-Shirshov basis theoremin the group. The thesis consists of three sections. In the first section we mainly introducethe definition of Grobner-Shirshov basis and relative lemmas. The second section give theapplication of Grobner-Shirshov basis in the free products with amalgamation and thealternating group. In the third section we generalize the Shirshov's Composition Lemmaby replacing the monomial order for a non-monomial order. From this result, we show bydirect calculations of compositions, that the new presentation of the HNN extension is aGrobner-Shirshov basis under an appropriate ordering of group words, in which the orderis not monomial order. By the generalized composition lemma, we immediately obtainthe Normal Form Theorem for HNN extensions.