Page 66 - DMTH403_ABSTRACT_ALGEBRA

Unit 4: Lagrange's Theorem

4.4 Keywords Notes

Coset: Let H be a subgroup of a group G, and let x  G. We call the set Hx = {hx | h  H} a right
coset of H in G.
Lagrange: Let H be a subgroup of a finite group G. Then o(G) = o(H) | G : H |. Thus, o(H) divides
o(G) and | G : H | divides o(G).
4.5 Review Questions

1. Obtain the left and right cosets of H = < (1 2) > in S . Show that Hx  xH for some x  S . 3
3
2. Show that K = {I, –I} is a subgroup of Q . Obtain all its right cosets in Q .
8
8
3. Let H be a subgroup of a group G. Show that there is a one-to-one correspondence between
the elements of H and those of any right or left coset of H.
(Hint: Show that the mapping f : H  Hx : f(h) = hx is a bijection.)
4. Write Z as a union of disjoint cosets of 5Z.

5. Check that f is a bijection.
6. What are the orders of
(a) (1 2)  S , (b) I  S ,
3
4
0  1
(c)    Q , (d) 3  Z 4
8
  1 0 
(e) 1  R?

Answers: Self Assessment

1. (b) 2. (b) 3. (d) 4. (c) 5. (a)

4.6 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/
archives.math.utk.edu

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