List the Sylow 2subgroups Sylow 3subgroups and Sylow 5subgro

List the Sylow 2-subgroups, Sylow 3-subgroups, and Sylow 5-subgroups of Z_12 times Z_12 times Z_10+ [Section 9.2 is a prerequisite for this exercise.]

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In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.

For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e. a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Syl_p(G).

The Sylow theorems assert a partial converse to Lagrange\'s theorem. While Lagrange\'s theorem states that for any finite group G the order (number of elements) of every subgroup of G divides the order of G, the Sylow theorems state that for every prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G. The order of a Sylow p-subgroup of a finite group G is pⁿ, where n is the multiplicity of p in the order of G, and every subgroup of order pⁿ is a Sylow p-subgroup of G. The Sylow p-subgroups of a group (for a given prime p) are conjugate to each other. The number of Sylow p-subgroups of a group for a given prime p is congruent to 1 mod p.

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Theorems

Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of Syl_p(G), all members are actually isomorphic to each other and have the largest possible order: if |G| = pⁿm with n > 0 where p does not divide m, then every Sylow p-subgroup P has order |P| = pⁿ. That is, P is a p-group and gcd(|G : P|, p) = 1. These properties can be exploited to further analyze the structure of G.

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.

Theorem 1: For every prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order pⁿ.

The following weaker version of theorem 1 was first proved by Cauchy, and is known as Cauchy\'s theorem.

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element (and hence a subgroup) of order p in G.^[1]

Theorem 2: Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate to each other, i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g¹Hg = K.

Theorem 3: Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pⁿm, where n > 0 and p does not divide m. Let n_p be the number of Sylow p-subgroups of G. Then the following hold:

Consequences

The Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, pⁿ. Conversely, if a subgroup has order pⁿ, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pⁿ.

A very important consequence of Theorem 2 is that the condition n_p = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup (there are groups that have normal subgroups but no normal Sylow subgroups, such as S₄).