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### 25 Cards in this Set

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 Define what a normal subgroup is A subgroup H of a group G is called a normal subgroup of G if aH=Ha for all a in G What is the normal subgroup test? A subgroup H of G is normal in G if and only if xHx^(-1) is contained in H for all x in G What are some examples of groups that we are familiar with that are always normal? (a) Every subgroup of an Abelian group is Normal (b) The center of a group is always normal (c) The alternating group An of even permutations is a normal subgroup of Sn What is the significance of normal groups? If a subgroup H is normal, then the set of left (or right) cosets of H in G is itself a group (called the factor or quotient group) If G is a group and H is a normal subgroup of G, how do we define the factor group G/H? The set G/H= { aH : a R(g) is an isomorphism Corollary to First Isomorphism Theorem If R is a homomorphism from a finite group G to G', then |R(G)| divides |G| and |G'| Every normal subgroup of a group is... ...the kernel of a homomorphism. In particular, a normal subgroup N is the kernel of the mapping g ---> gN from G to G/N State the Fundamental Theorem of Finite Abelian Groups Every finite Abelian group is a direct product of cyclic groups of prime-power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group If m divides the order of a finite Abelian group... ...then G has a subgroup of order m