Group Theory Theorems and Definitions
Explore essential theorems in group theory including subgroup criteria, intersection properties, and the center of a group definition. Learn about subgroup intersections, abelian groups, and the center of a group.
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Presentation Transcript
theorem Theorem(1): Anon-empty subset H of G is subgroup of the group (G,*) if satisfy the following condition 1- for each a,b in H; a*b H 2- a^-1 H for each a in H
Theorem (2) Let (G,*)be agroup & ? H G then (H,*) is subgroup of (G,*) iff a*(b)^-1 H for each a,b in H
Theorem 3 The intersection of two subgroup of (G,*) is subgroup of (G,*) Theorem 4: If (H,*) and (K,*) are two subgroup of (G,*) THEN (H ?,*)subgroup of (G,*) iff H K v K H
Center of group Def: if (G,*) is agp the center of G is denoted by Cen(G) =Z(G) and defined as Z(G)={x G;x*a=a*x for each a G} G EX : (z,+) is agp cent(Z)=Z
theorem Agrup (G,*) is abelain iff cent(G)=G theorem 2) for any gp (G,*) (Z(G),*) Is subgroub of (G,*)
