Tag Archives: proof

The chocolate bar game: losing positions proved

In my last post I claimed that the losing positions for the chocolate bar game are precisely those of the form (or the reverse), that is, in binary, positions where one coordinate is the same as the other with any … Continue reading

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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

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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

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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

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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

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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

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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

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