Subgroup (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-07

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Subgroup


00:00:41 1 Basic properties of subgroups
00:01:22 2 Cosets and Lagrange's theorem
00:01:43 3 Example: Subgroups of Zsub8/sub
00:02:24 4 Example: Subgroups of Ssub4 /sub(the symmetric group on 4 elements)
00:02:45 4.1 12 elements
00:03:26 4.2 8 elements
00:04:07 4.3 6 elements
00:04:48 4.4 4 elements
00:05:30 4.5 3 elements
00:05:50 5 Other examples
00:06:11 6 See also
00:06:32 7 Notes
00:06:52 8 References
00:07:13 Other examples
00:07:54 See also



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SUMMARY
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In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H  G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).If H is a subgroup of G, then G is sometimes called an overgroup of H.
The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.