Math 436 Notes: Series
نویسنده
چکیده
Definition 1.1 (Series). Fix a group G and subgroups H ≤ K ≤ G. An ascending series connecting H and K, Ĝ∗ ↑ K H is a sequence of groups of the form: H = G0 E G1 E · · · E Gn−1 E Gn = K. Note it is only assumed that Gi is normal in Gi+1 and not neccesarily in G. If Gi E G for all i we call Ĝ∗ ↑ K H a normal series. A descending series connecting K and H, Ĝ∗ ↓ K H is a sequence of groups of the form: H = Gn E Gn−1 E · · · E G1 E G0 = K A normal descending series is a series where Gi E G for all i. Notice we may reindex a series by i ↔ n− i to change from a descending series to an ascending one and vice versa. For a finite series, we will often write Ĝ∗| K H if we do not want to emphasize the indexing. If we refer to Ĝ∗ as a series for G without specifying H and K it is understood that Ĝ∗ = Ĝ∗| G e , i.e., that H = e and K = G. We also will consider infinite series on occasion: An infinite ascending series Ĝ∗ ↑H is one of the form: H = G0 E G1 E G2 E . . . . Similarly an infinite descending series Ĝ∗ ↓ K is one of the form: · · ·E G2 E G1 E G0 = K.
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