View source for Simple group

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A group without normal subgroups different from the unit subgroup and the entire group (cf. [[Normal subgroup|Normal subgroup]]). The description of all finite simple groups is a central problem in the theory of finite groups (cf. [[Simple finite group|Simple finite group]]). In the theory of infinite groups the significance of simple groups is substantially less, as they are difficult to visualize. The group of all even permutations fixing all but a finite number of elements of a set $M$ is simple if $M$ has cardinality at least 5. If $M$ is infinite, this group is infinite too. There exist finitely-generated, and even finitely-presented, infinite simple groups. Every group can be imbedded in a simple group. The definition of a simple group given here differs somewhat from that given in the theory of Lie groups and algebraic groups (cf. [[Lie group, semi-simple|Lie group, semi-simple]]).



====Comments====
In the theory of infinite groups two notions stronger than simplicity are used, viz. those of an absolutely simple group and a strictly simple group. One has the implications: absolutely simple $\Rightarrow$ strictly simple $\Rightarrow$ simple. There are examples of simple groups that are not absolutely simple and of simple groups that are not strictly simple.

A group is strictly simple if it has no non-trivial ascendent subgroup; it is absolutely simple if it has no non-trivial [[Serial subgroup|serial subgroup]]. Cf. [[#References|[a6]]] for more details.

An algebraic group over an algebraically closed field is simple if it has no closed non-trivial normal subgroup. It is quasi-simple, or almost simple, if it has no non-trivial infinite normal subgroup. If $G$ is almost simple, then the abstract group $G/Z(G)$, where $Z(G)$ is the centre, is simple as an abstract group.

A Lie group is simple if it has no non-trivial Lie subgroup. For a connected Lie group this is the same as simplicity of its Lie algebra.

A topological group is called simple if it has no proper closed normal subgroup.

Both for algebraic groups and topological groups one also finds in the literature the definition that such a group is simple if it has no non-trivial closed connected normal subgroup.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1939) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.W. Mackey, "Unitary group representations" , Benjamin/Cummings (1978) {{MR|0515581}} {{ZBL|0401.22001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Freudenthal, H. de Vries, "Linear Lie groups" , Acad. Press (1969) {{MR|0260926}} {{ZBL|0377.22001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Weinstein, "Examples of groups" , Polygonal (1977) pp. Examples 6.14; 6.15 {{MR|0453847}} {{ZBL|0359.20001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , '''1''' , Springer (1972) pp. Chapt. 1 {{MR|0332990}} {{MR|0332989}} {{ZBL|}} </TD></TR></table>