Application of Group Theory in Mathematics

650 Words Feb 5th, 2018 3 Pages
The most basic forms of mathematical groups are comprised of two group theory elements which are combined with an operation and determined to equal a third group element (Baumslag, 1999). When group theory is applied, then many mathematical patterns can be defined and solved including rings and topological spaces. Within this theory there are four basic rules that must be followed when determining groups. The first rule is known as closure. The closure rule states that, "If a and B are in a group then a x b is also in that group." (Novikov, 1955). In other words, in order for their to be a valid group, then the binary operation, here the "x" symbol, has to allow for the answer to match the other two groups. So, if the first group is -6 and the second group is -3, then only the addition and subtraction binary operations can be used, because the answer must also be negative (Nathanson, 2010). If the multiplication or division binary operation was used, then the answer would be positive and would not match the rest of group, causing their to be no closure. The second rule is the rule of associativity. The rule of associativity states that, "If a,b, and c are in the group then (a * b) * c = a * (b * c)." (Novikov, 1955). In other words, the group must have the same answer regardless of the…
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