The Church of Jesus Christ of Latter-day Saints, a.k.a. Mormon -- does not authorize the practice of polygamy. Polygamy in the United States is illegal. Any member found doing so would be immediately excommunicated from the Church. There are no small break-off groups that are recognized by the Church that practice polygamy. That is a false assumption.
Again, let me emphasize clearly -- that no active members in good standing with the LDS Church, anywhere in the world, practice polygamy.
The show "Big Love" give a completely distorted view of Mormonism and its teachings about marriage, etc... Any information that you get about Mormons, from that show -- is most likely completely incorrect. And certainly, sensational.
"The family is ordained of God. Marriage between man and woman is essential to His eternal plan. At certain times and for His specific purposes, God, through His prophets, has directed the practice of plural marriage (sometimes called polygamy), which means one man having more than one living wife at the same time. In obedience to direction from God, Latter-day Saints followed this practice for about 50 years during the 1800s but officially ceased the practice of such marriages after the Manifesto was issued by President Woodruff in 1890. Since that time, plural marriage has not been approved by The Church of Jesus Christ of Latter-day Saints and any member adopting this practice is subject to losing his or her membership in the Church." LDS.org