(a) а aH (b) aH = H if and only if a e H (c) aH = bH or aHn bH = ø (d) aH = bH if and only if a ' b e H %3D (e) |aH| = |bH| = |H| %3D (f) aH = Ha if and only if H = a 1 Ha (a) aH is a subgroun of G if and only if a e H

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Cosets
Let H be a subgroup of G and let a and b belong to G
Then the following conditions hold
Proof:

(a)
a e aH
(b)
aH = H if and only if a e H
(c)
aH = bH or aHn bH = Ø
(d)
aH = bH if and only if a be H
(e)
|aH| = |bH| = |H|
(f)
aH = Ha if and only if H = a1 Ha
(g)
aH is a subgroup of G if and only if a e H
Transcribed Image Text:(a) a e aH (b) aH = H if and only if a e H (c) aH = bH or aHn bH = Ø (d) aH = bH if and only if a be H (e) |aH| = |bH| = |H| (f) aH = Ha if and only if H = a1 Ha (g) aH is a subgroup of G if and only if a e H
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