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# Tag Archives: subgroups

## MaBloWriMo 30: Cyclic subgroups

Today, to wrap things up, we will use Lagrange’s Theorem to prove that if is an element of the group , the order of evenly divides the order of . So we have a group and an element . In … Continue reading

Posted in algebra, group theory, proof
Tagged cyclic, groups, Lagrange, MaBloWriMo, proof, subgroups
6 Comments

## MaBloWriMo 27: From subgroups to equivalence relations

Again, let be a group and a subgroup of . Then we can define a binary relation on elements of , called , as follows: if and only if there is some such that . That is, for any two … Continue reading

Posted in algebra, group theory, proof
Tagged equivalence, groups, Lagrange, MaBloWriMo, proof, relation, subgroups

## MaBloWriMo 26: Left cosets

Let be a group and a subgroup of . Then for each element we can define a left coset of by . That is, is the set we get by combining (on the left) with every element of . For … Continue reading

Posted in algebra, group theory, proof
Tagged cosets, groups, Lagrange, MaBloWriMo, proof, subgroups
1 Comment

## MaBloWriMo 25: Subgroups

So in the remainder of the month, we’ll prove that in any group , the order of each element must evenly divide the order (size) of the group. I said in an earlier post that this is called Lagrange’s Theorem; … Continue reading

Posted in algebra, group theory, proof
Tagged groups, Lagrange, MaBloWriMo, proof, subgroups
1 Comment