Infinite Index
If H has an infinite number of cosets in G, then the index of H in G is said to be infinite. In this case, the index |G : H| is actually a cardinal number. For example, the index of H in G may be countable or uncountable, depending on whether H has a countable number of cosets in G. Note that the index of H is at most the order of G, which is realized for the trivial subgroup, or in fact any subgroup H of infinite cardinality less than that of G.
Read more about this topic: Index Of A Subgroup
Famous quotes containing the words infinite and/or index:
“The religion of the Bible is the best in the world. I see the infinite value of religion. Let it be always encouraged. A world of superstition and folly have grown up around its forms and ceremonies. But the truth in it is one of the deep sentiments in human nature.”
—Rutherford Birchard Hayes (1822–1893)
“Exile as a mode of genius no longer exists; in place of Joyce we have the fragments of work appearing in Index on Censorship.”
—Nadine Gordimer (b. 1923)