Prime power order implies subgroups of all orders dividing the group order
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group of prime power order) must also satisfy the second group property (i.e., group having subgroups of all orders dividing the group order)
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Suppose is a group of prime power order . Suppose . Then, has a subgroup of order .
Congruence condition on the number of subgroups
- Congruence condition on number of subgroups of given prime power order: This states that for any finite group , and any prime power dividing the order of , the number of subgroups of of order is congruent to modulo . The special case of this (which is used as a first step in the proof) is where the order of is itself a power of .
- Finite group has subgroups of all prime power orders dividing its order
- Finite nilpotent iff normal subgroups of all orders dividing the group order
- Subgroups of all orders dividing the group order implies solvable
Given: A group of prime power order . An integer with .
To prove: has a subgroup of order .
Proof by moving upward on
Proof: We prove the result by induction on . The base case, , is trivial. Suppose . Note also that if , the trivial subgroup works, so we can consider .
- has a normal subgroup of order : Since is nontrivial, fact (1) tells us that its center is nontrivial. Pick any non-identity element . The order of is , with <mah>1 \le j \le n</math>. Pick . has order , and since , is normal.
- has a subgroup of order : has order , so the induction hypothesis applies to yield that it has a subgroup, say , of order .
- The subgroup has order : This is a direct consequence of Lagrange's theorem, and what we know about the orders of and .