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Tag Archives: Lagrange
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 29: Equivalence classes are cosets
Today will conclude the proof of Lagrange’s Theorem! Recall that we defined subgroups and left cosets, and defined a certain equivalence relation on a group in terms of a subgroup . Today we’re going to show that the equivalence classes … Continue reading
Posted in algebra, group theory, proof
Tagged classes, cosets, equivalence, groups, Lagrange, MaBloWriMo, proof
Comments Off on MaBloWriMo 29: Equivalence classes are cosets
MaBloWriMo 28: Equivalence relations are partitions
Today we’ll take a brief break from group theory to prove a fact about equivalence relations, namely, that they are the same as partitions. A partition is a pretty intuitive concept: you take a big set, and cut it up … Continue reading
Posted in algebra, group theory, proof
Tagged equivalence, groups, Lagrange, MaBloWriMo, partition, proof, relation
2 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
Comments Off on MaBloWriMo 27: From subgroups to equivalence relations
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